Fracture Mechanics of OneDimensional Nanostructures
Northwestern University
Rod Ruoff
Nanotechnology
Carbon Nanotube Source
Arc-grown Multi-wall Carbon Nanotubes (MWCNTs) from MER Corp.
AZ. were studied in this work. A simple fractionation process was used
to remove some impurities and increase nanotube concentration.
SEM image of powdered cathode deposit
core material with 30-40% MWCNT
content from MER Corp.
Northwestern University
SEM image of separated MWCNTs on a
silicon wafer, after fractionation.
Rod Ruoff
Nanotechnology
Testing Tool: Nanomanipulator
A home-built nanomanipulator is used to perform mechanics study inside
vacuum chamber of an scanning electron microscope (SEM).
Piezoelectric Actuator
X-Y Stage
Z-stage
Nanomanipulator inside vacuum chamber
of FEI Nova 600 SEM (Ruoff group)
Home-built nano-manipulator
X-Y
stage
Piezo
bimorph
Northwestern University
Z stage
Cantilever
Holder
Specimen/
Cantilever
Holder
Rod Ruoff
Nanotechnology
Nanoscale Tensile Test
Atomic force microscope (AFM) cantilevers are used as manipulation tools and
force-sensing elements.
Tensile Test Schematic
Experimental Setup
X-Y Stage
soft cantilever
L
rigid cantilever
Z Stage
L+
MWCNT
Northwestern University
s
Rod Ruoff
Nanotechnology
In situ Clamping - EBID
Electron beam induced deposition (EBID) is the process of using a highintensity electron beam to deposit structures on a scanned surface. EBID
is commonly used to make clamps in situ inside SEM.
EBID clamp
Hydrocarbon
molecules
Exposure area
EBID principle
Northwestern University
A CNT in contact with an AFM tip, before and
after EBID clamping
Rod Ruoff
Nanotechnology
“Sword-in-sheath” Fracture
Multi-wall carbon nanotubes fracture in a “sword-in sheath” manner during
tensile test.
Cross-sectional area:
Inner shells
A  πDΔ
Outer shell
D

: inter-layer separation of graphite, 0.34 nm
Northwestern University
Rod Ruoff
Nanotechnology
MWCNT Diameter Measurement
Cantilever holders were designed to hold a shortened AFM chip for nanotube
diameter measurement in TEM.
AFM
cantilevers
(a) AFM chip holder model
(b) An AFM chip in the AFM chip holder
Northwestern University
(c) Gatan TEM straining holder (model 654)
(d) SEM and TEM images of a MWCNT fragment
attached to an AFM tip.
Rod Ruoff
Nanotechnology
Stress & Strain Measurements
The whole tensile testing process was recorded by taking SEM
images at each loading step.
Northwestern University
Rod Ruoff
Nanotechnology
MWCNT Tensile Testing Result
Fracture Strength
Elastic Modulus
Average elastic modulus: ~ 910 GPa
Northwestern University
Rod Ruoff
Nanotechnology
Multiple Loading (Tube #6)
(1)
(2)
(3)
Test #
Gauge
Length
(m)
Breaking
Force
(nN)
Tensile
Strength
(GPa)
Failure
Strain
(%)
Elastic
Modulus
(GPa)
1
4.08
220
21  1.4
1.8  0.3
1200  210
2
3.75
240
23  1.4
1.9  0.3
1250  210
3
3.46
420
41  2.6
3.0  0.2
1230  130
Northwestern University
Rod Ruoff
Nanotechnology
Nanoparticle Chain Aggregates
Stretching a chain
Carbon nanoparticle chain aggregates
Tensile Testing
Contact Force Measurement
• Nanoparticle diameter: 25-35 nm
• Chain length: ~ 2 m
• Breaking Force: 42  25 nN
• Tensile Strength: 40 -100 MPa
• Elastic Modulus : 200 - 600 MPa
• Particle Contact Force: 8  4 nN
Northwestern University
Rod Ruoff
Nanotechnology
Nonlinearity: Large Deflection + Misalignment
L
EI
x
x
(x,y)
s
d
 M  Fy ( L   x  x)  Fx ( y  y )
ds
where Fx=Fsin and Fy=Fcos
y

0
Fx
F
EI 0
1
(
d ) 2
2 0
2L
sin o     sin    
y
 y   dy 
0
y
Fy
F
EI
2F
 x L 
EI
2F
0

0
0

0
sin 
d
sin(0   )  sin(   )
cos
d
sin(0   )  sin(   )
(Transformation; Converting to
elliptic integrals)
EI
( F (k )  F (1 , k ))2
2
L


E (k )  E (1 , k )
2k cos1
 y  (1  2
) cos 
sin   L
F (k )  F (1 , k )
F (k )  F (1 , k )


F

 x  1  (1  2


E (k )  E (1 , k )
2k cos1
) sin  
cos  L
F (k )  F (1 , k )
F (k )  F (1 , k )

F(k),F(,k) complete and incomplete elliptic integral of first kind. k and 1 are
obtained from angle 0 with following relationships:
sin 0  2k 2  1
Northwestern University
Rod Ruoff
sin 1 
2
2k
Nanotechnology
Experimental Data Analysis
• AFM cantilevers were used as force-sensing elements in our nanoscale
tensile testing experiments on templated carbon nanotubes inside SEM.
• Large deflection of the cantilever beam was encountered in the tests along
with non-ideal alignment of the specimens.
• The linear analysis underestimated the applied load up to 15 %.
Slope
0
(degree)
Loading
Angle

(degree)
Vertical Deflection y
Applied Load
Linear
(N)
Analytical
(N)
Error
(%)
Measured
(m)
Linear
(m)
Analytical
(m)
11.0
36.8
25.3
29.7
14.8
42.9
42.2
42.3
17.4
5.5
23.5
24.9
5.6
68.7
70.8
70.3
9.22
42.0
20.8
23.9
13.0
30.6
32.2
32.3
11.6
2.0
19.4
19.6
1.0
41.4
40.5
40.4
15.0
24.3
27.6
31.4
12.1
52.9
52.4
52.3
Northwestern University
Rod Ruoff
Nanotechnology
Error in Linear Estimation
Normalization
Linear analysis
FL  K yL / cos  2
EI0
2
L cos

FL L2
2 0
EI
cos
 yL
2
3
2
 0
L
3
 yL  0 L
Analytical analysis
F
EI 0
1
(
d ) 2
2 0
2L
sin(o   )  sin(   )
EI
( F (k )  F (1 , k ))2
2
L
y
EI 0
sin 
 y   dy 
d

0
2 F 0 sin(0   )  sin(   )



E (k )  E (1 , k )
2k cos1
 (1  2
) cos 
sin  L
F (k )  F (1 , k )
F (k )  F (1 , k )


FA L2
 ( F (k )  F (1 , k ))2
EI
 yA
E (k )  E (1 , k )
2k cos1
 (1  2
) cos 
sin 
L
F (k )  F (1 , k )
F (k )  F (1 , k )
Northwestern University
Rod Ruoff
Nanotechnology