Supporting Information:
Exceptional resilience of small-scale Au
30
Cu
25
Zn
45
under cyclic
stress-induced phase transformation
Xiaoyue Ni
1
, Julia R. Greer
1
, Kaushik Bhattacharya
1
, Richard D. James
2
, and Xian Chen
*
3
1
Division of Engineering and Applied Science, California Institute of Technology, Pasadena CA
91125
2
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis,
Minnesota 55455
3
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and
Technology, Hong Kong
TABLE OF CONTENTS
I. Information of supporting movies
· · · · · · · · · · · · · · · · · · · · · ·
1
II. Geometrically nonlinear theory of martensite for the formation of microstructure un-
der uniaxial compression
· · · · · · · · · · · · · · · · · · · · · · · · · · ·
2
II.1
Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
II.2
Crystallography
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
II.3
Energy minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
II.4
Deformation by forming (
̄
101) type I twin . . . . . . . . . . . . . . . . . . . . . . . .
6
III. Reference figure for the re-entrant corner of martensite microstructure
· · · · · ·
7
I. Information of supporting movies
Movie S1: Evolution of microstructure of Au
30
Cu
25
Zn
45
1
μ
m-diameter pillar during
in-situ
compression-induced martensitic transformation
Movie S2: Evolution of microstructure of Au
30
Cu
25
Zn
45
2
μ
m-diameter pillar during
in-situ
compression-induced martensitic transformation
The movies record the morphology change and the corresponding stress-strain response
(converted from the force-displacement data) of the Au
30
Cu
25
Zn
45
pillars within a complete
*
To whom all correspondences should be addressed. Email: xianchen@ust.hk
cycle of phase transformation.
The nanomechanical experiments were conducted by a
custom-made
in-situ
nanoindenter module (PI 85 PicoIndentor, Hystron, Inc) inside a
scanning electron microscope (Versa 3D, FEI).
II. Geometrically nonlinear theory of martensite for the formation of
microstructure under uniaxial compression
In this section of the SI, we will give a mathematical statement for determination of martensite
variant that will form in favor of the uniaxial compressive loading condition.
II.1 Kinematics
Consider a Au
30
Cu
25
Zn
45
pillar defined in Figure S1 subjected to
w
:
h
=
1 : 3 as caved from a
single grain of austenite before loading.
Upon loading, the cubic to monoclinic phase
transformation will be induced by the uniaxial compressive stress. Such a symmetry-breaking
transformation results in a set of 12 distinct martensite variants
13
:
M
=
{
U
1
, ...,
U
12
} yielding
the following order,
U
1
=
α≤
0
≤δ
0
0
0
γ
,
U
2
=
α
−
≤
0
−
≤δ
0
0
0
γ
,
U
3
=
δ≤
0
≤α
0
0
0
γ
,
U
4
=
γ
0
0
0
δ
−
≤
0
−
≤α
(S1)
U
5
=
γ
0
0
0
δ≤
0
≤α
,
U
6
=
α
0
−
≤
0
γ
0
−
≤
0
δ
,
U
7
=
α
0
≤
0
γ
0
≤
0
δ
,
U
8
=
δ
0
≤
0
γ
0
≤
0
α
(S2)
U
9
=
γ
0
0
0
α≤
0
≤δ
,
U
10
=
γ
0
0
0
α
−
≤
0
−
≤δ
,
U
11
=
δ
0
−
≤
0
γ
0
−
≤
0
α
,
U
12
=
δ
−
≤
0
−
≤α
0
0
0
γ
. (S3)
The values
α
=
1.0591,
≤
=
0.0073,
δ
=
1.0015 and
γ
=
0.9363 are determined by the lattice
parameters of austenite and martensite, which were measured by the X-ray diffraction
experiment
12
. The blue domain
Ω
M
in Figure S1 is considered as a mixture of deformations of
martensite due to the structural phase transformation. Let
f
∈
[0, 1] represent the average
volume fraction for twinning, the average deformation gradient within the martensite region
can be interpreted as
F
f
=
ˆ
R
f
[(1
−
f
)
U
+
f
ˆ
U
],
(S4)
for
U
,
ˆ
U
∈
M
related by
ˆ
U
=
(
−
I
+
2
e
⊗
e
)
U
(
−
I
+
2
e
⊗
e
) where
e
is one of the two-fold axes of
austenite,
|
e
| =
1.
ˆ
R
f
∈
SO
(3) is some rigid rotation depending on
f
. The austenite and
martensite interface
m
(also called the habit plane by metallurgists), can be determined from
the crystallographic equation
27
F
f
−
I
=
b
⊗
m
,
(S5)
for some vector
b
∈
R
3
. This equation is not generally solvable due to the lack of compatibility
between the two lattices. Once the cofactor conditions are satisfied, we can find a pair of solu-
tions to the Eq. (S5) for every
f
∈
[0, 1]
13
. In particular,
f
=
0 and
f
=
1 correspond to the habit
plane between a single martensite variant and austenite.
ΩΩ
A
Ω
A
Ω
M
F
f
m
f
-
ˆ
N
Figure S1: Kinematic model for microstructure evolution from austenite to martensite subjected to uni-
axial compression.
II.2 Crystallography
We utilized the X-ray Laue microdiffraction to characterize the crystallographic orientation of
the Au
30
Cu
25
Zn
45
pillars used in our nanomechanical experiments. This measurement was
conducted at the Advanced Light Source Beamline 12.3.2. An area of 3
×
4 mm
2
was scanned
by the polychromatic synchrotron X-rays from which we collected a sequence of Laue patterns
to construct the orientation mapping for the sample surface by the method outlined in
reference
15
. Figure S2(a) shows the spatial distribution of angle between Z-axis and c-axis
where X-Y-Z and a-b-c represent the sample and crystal coordinates respectively as illustrated
in Figure S2(c). It shows an angular variation of 2 degree from 10.6 to 12.4 within the scanned
austenite region. The region chosen for FIBing is marked as the black box in Figure S2(a),
which corresponds to the SEM image in Figure S2(b). The Laue pattern of austenite within this
region is shown in Figure S2(d) and the angle between the normal of sample surface and the
c-axis is 11.25 degree. The outer normal vector of these micro-pillars is characterized as
ˆ
N
=
1
ρ
N
O
A
[001]
L
2
1
=
(0.150,
−
0.125, 0.981) for some scalar
ρ
N
,
(S6)
in which the orientation tensor
O
A
=
1.6039
−
0.05659
0.2431
0.08753
1.6081
−
0.2032
−
0.2338
0.21104
1.59199
.
(S7)
Figure S2: Orientation information of the austenite domain from which the micro-pillars were fabri-
cated. (a) Orientation mapping of the area scanned by the polychromatic synchrotron X-ray. (b) The
overview of SEM image of pillars within the region marked in (a). (c) The orientation of the sample coor-
dinate, X-Y-Z, relative to the crystal coordinate of austenite, a-b-c, corresponding to (d) Laue pattern.
II.3 Energy minimization
In the scenario that the reference configuration in austenite phase is deformed by a uniaxial
loading, the total free energy
E
depends on the deformation gradient
∇
y
, where
y
:
Ω
→
R
3
,
E
=
∫
Ω
(
φ
(
∇
y
)
−
P
:
∇
y
)
dx
.
(S8)
The first term of integrand is the Helmholtz free energy density.
P
is the first Piola-Kirchhoff
stress defined as
P
=
σ
ˆ
N
⊗
ˆ
N
(S9)
where
σ
<
0 for compression and
ˆ
N
is the outer normal of top surface of the pillar, measured in
previous section. We need to find a deformation gradient
F
∈
R
3
×
3
that minimizes
φ
(
F
)
−
P
:
F
,
and the minimizing deformation will be given by
y
=
Fx
for
x
∈
Ω
.
For the martensitic transformation, we have the martensite energy wells
W
M
=
{
SO
(3)
U
:
U
∈
M
} such that
φ
(
F
)
<
φ
(
A
) for all
A
∈
R
3
×
3
if
F
∈
W
M
. Thus, we can find a martensite variant from
the set
M
that minimizes Eq. (S8) with the application of the load given in Eq. (S9) by solving
the following maximization problem:
max
U
∈
M
(
max
R
∈
SO
(3)
σ
U
ˆ
N
·
R
T
ˆ
N
)
.
(S10)
Since
σ
<
0, this maximization problem can be converted to a minimization problem
28
:
min
U
∈
M
(
√
ˆ
N
·
U
2
ˆ
N
)
,
(S11)
subjected to some constraint on rotation matrix
R
to avoid the case that
R
ˆ
N
=−
ˆ
N
. By direct
calculation, (S11) is minimized by four martensite variants and they are
U
1
,
U
2
,
U
3
and
U
12
listed in (S1)-(S3). The uniaxial recoverable strain is calculated by
≤
N
=
√
ˆ
N
·
U
2
ˆ
N
−
1
=−
0.047
(S12)
and the shear strain is determined by
≤
S
=
tan
(
arccos
ˆ
N
·
F
ˆ
N
|
F
ˆ
N
|
)
,
(S13)
for
F
=
ˆ
RU
that solves the crystallographic equation (S5). In the case of
U
1
, there exist two
conjugate solutions for
ˆ
R
, vector
b
and the habit plane
m
listed in Table S1, which result in two
shear strains, 0.0699 and 0.0395.
ˆ
R
b
m
Sol. I
0.9985
−
0.000182
0.05428
−
0.000182
0.99998
0.006705
−
0.05428
−
0.006705
0.9985
−
0.07758
−
0.00958
0.07758
0.7416
0.0916
0.6645
Sol. II
0.9985
−
0.000182
−
0.05428
−
0.000182
0.99998
−
0.006705
0.05428
0.006705
0.9985
0.07758
0.00958
0.07758
0.7416
0.0916
−
0.6645
Table S1: Two solutions of the crystallographic equation (S5) for variant
U
1
.
II.4 Deformation by forming
(
̄
101)
type I twin
We construct a set of twin lamellae by the variant
U
1
and variant
U
5
listed in (S1) and (S2),
which satisfy the cofactor conditions for (
̄
101) type I twin closely (verified in reference
15
). A
deformation sequence is proposed as
y
(
f
;
x
)
=
F
f
x
+
a
∫
(
x
·
n
)
0
χ
(
f
;
s
)d
s
,
if
g
(
f
;
x
)
>
0
x
,
else.
(S14)
The deformation gradient
F
f
is calculated from Eq.
(S4) by substituting
U
=
U
1
and
e
=
1
p
2
(
̄
101) for the volume fraction
f
∈
{0.0, 0.45, 0.05, 0.46, 0.15, 0.47, 0.15, 0.05, 0.05, 1.0}. The
resultant normal strain is -0.03207 by Eq. (S12) and the shear strain is 0.01885 by Eq. (S13). The
vectors
a
=
[0.1624, 0.02, 0.1455] and
n
=
e
are the type I twinning parameters.
χ
(
f
;
s
) is a
piecewise function defined as,
χ
(
f
;
s
)
=
(1
−
f
)(
s
−
b
s
c
),
b
s
c
≤
s
<
b
s
c
+
f
;
−
f
(
s
−
d
s
e
),
b
s
c
+
f
≤
s
<
d
s
e
.
(S15)
This function ensures a
rank-1
interface between the two martensite variants
13
. The function
g
(
f
;
x
) defines the austenite and twinned martensite interfaces consisting of alternative
normals
m
0
=
(0.7416, 0.0916,
−
0.6645) and
m
1
=
(
−
0.6645, 0.0916, 0.7416),
g
(
f
;
x
)
=
(
x
−
p
f
)
·
m
1
,
b
x
·
n
c
≤
x
·
n
<
b
x
·
n
c
+
f
(
x
−
p
f
)
·
m
0
,
b
x
·
n
c
+
f
≤
x
·
n
<
d
x
·
n
e
.
(S16)
The vector
p
f
is the vertex of a triple junction at which the martensite twin pair meet austenite
as shown in Figure S4.
Figure S3: Compatible interface between austenite and type I twinned martensite at volume fraction
f
=
0.4. The red region are the austenite (undeformed phase), while the blue/green lamellae are the twin
pair corresponding to (
U
1
,
U
5
). Either variant can form a exact interface with austenite lattice. Austenite
and two martensite variants meet at the triple junction
p
f
.
III. Reference figure for the re-entrant corner of martensite microstructure
Figure S4: Formation of the type II twin boundary in Cu-Al-Ni single crystal (in the paper by Ichinose et al
1985
25
). (a) - (c) shows the evolution of the twin boundary during stress-induced phase transformation.
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