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9.11.2015 PY5020 Nanoscience 59
Nanoindentation
References
• Nanoindentation, 2nd Ed., Anthony C. Fischer-Cripps, Springer, 2010.
• Introduction to Contact Mechanics, 2nd Ed., Anthony C. Fischer-Cripps,
Springer, 2007.
• Contact Mechanics, Kenneth L. Johnson, ,1985
• M. R. VanLandingham, Review of instrumented indentation, J. Res. Natl.
Inst. Stand. Technol. 108, 249-265 (2003).
• W. C. Oliver, G. M. Pharr, Measurement of hardness and elastic modulus
by instrumented indentation: Advances in understanding and refinements
to methodology, J. Mater. Res., 19(1) (2004).
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Nanoindentation
Touch the surface of a material to determine mechanical properties
Origins in Mohs’ hardness scale of 1822 based on scratching, numbered 1-10
Followed by Brinell, Rockwell, Vickers and Knoop indentation 1900-1935
Distinguishing features of nanoindentation:
• Displacement is continuously measured with resolution in nanometers
• Load is continuously measured in nanoNewton to milliNewtons
• Contact area is inferred from known geometry and properties of indenting
tip
• Small amplitude oscillatory response is continuously measured
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Nanoindentation
Properties measured:
• Hardness
• Elastic modulus (..Poisson’s ratio, compliance tensor,..)
• Strain hardening exponent
• Fracture toughness (brittle)
• Viscoelastic properties (Storage and Loss Modulus, complex viscosity,…)
• Adhesion properties (interfacial fracture toughness, …)
• Isolated micro/nanoscopic defect and defect population behaviour
• Coupled parameters: Raman signal, electrical, etc.
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Instrumentation
One dimensional simple harmonic
oscillator
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Basic Measurement
Warning on notation differences from contact mechanics conventions:• For load (ie. force) nanoindentation
sometimes uses L instead of P• For displacement, nanoindentation
usually uses h instead of d
Berkovich 3-sided pyramid
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Elastic-plastic Deformation
The elastic limit of deformation of most materials is a
few percent, then Hooke’s Law fails
Plasticity deals with behaviour beyond this limit.
Rigid plastic Elastic plastic Elastic strain
hardening plastic
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Yield Criterion
The physical conditions for yield in metals are:
Transition from elastic to plastic behaviour in ductile materials is called the
yield point.
Under the simple tension test, this occurs at the yield stress 0
• Hydrostatic pressure does not cause yield in solids (empirical result)
• For an isotropic material, the yield condition must be independent of
coordinate system
Yield conditions are based on invariants of the stress deviator
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Stress tensor decomposition
Define the mean normal stress
The stress tensor can be decomposed into a sum of a mean and deviation
A physical interpretation of this decomposition can be made when a body is
subject to an arbitrary stress:
0 0
0 0
0 0
xx xy xz m xx m xy xz
ij xy yy yz m xy yy m yz
xz yz zz m xz yz zz m
• The mean or hydrostatic component is associated with volume change
ie. Dilation or compression with shape self-similarity
• The deviatoric component is associated with shape change while preserving
volume
3
xx yy zz
m
Hydrostatic (H) Deviatoric (J)
By definition, a fluid is a body that, at rest, cannot sustain a
deviatoric component of stress (this happens at long times)
Total (I)
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Decomposition of stress
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Principle stresses and von Mises stress
In 3D, the following comprise the stress invariants. Ie. Quantities invariant under
coordinate transformation.
1 xx yy zzI 2 2 2
2 xx yy yy zz zz xx xy yz zxI
2 2 2
3 2xx yy zz xx yz yy zx zz xy xy yz zxI
von Mises Stress: The condition for plastic yield based on strain energy density
211 1 2
23 cos
3 3
II I
212 1 2
2 23 cos
3 3 3
II I
213 1 2
2 43 cos
3 3 3
II I
3
1 1 2 3
3 22
1 2
2 9 271arccos
3 2 3
I I I I
I I
The principle stresses in 3D are
1
2
3
0 0
0 0
0 0
ij
See Barber
2 2 2 2
1 2 1 2 2 3 3 13 ( ) ( ) ( )E I I
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Von Mises (Distortion Energy) Criterion
The Von Mises condition states that yield occurs when the second invariant of the
stress deviator J reaches a critical value k2 (eg. see Dieter p. 76):
2
2J k
1 0 2 3; 0 2 2
06 2k
To relate this physical condition to the stress in a simple tension test consisting
of uniaxial tension (assume we are aligned to principal directions for simplicity):
2 2 2
0 1 2 2 3 3 1
1( ) ( ) ( )
2
where2 2 2
2 1 2 2 3 3 1
1( ) ( ) ( )
6J
0 3k
2 2 2 2 2 2
0
1( ) ( ) ( ) 6( )
2xx yy yy zz zz xx xy yz zx
Or for an arbitrary coordinate system this becomes:
“The uniaxial yield stress or yield stress in tension”
For an arbitrary stress, in terms of this uniaxial yield stress we have:
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Distortion Strain Energy
1 1( )( )
2 2t xx xxU Pdu A e dx
For an elemental cube under tension
Elastic strain energy by work-energy theorem
Elastic strain energy density (per unit volume) is
1
2t tU Ph
1 1( )( ) ( )
2 2xx xx xx xxe Adx e dV
22
0
1 1 1( )
2 2 2
xxxx xx xxU e e E
E
2
2
0
1 1 1( )
2 2 2
xy
xy xy xyU e e
For shear:
0
1( )
2ij ijU e
2 2 2
0
1 1( ) ( ) ( )
2 2xx yy zz xx yy yy zz zz xx xy yz zxU
E E
In general:
(NB. Poisson effect does not appear as no extra lateral force applied)
(use simple case of P~h)
Peak load, displace.
Applying Hooke’s law
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Distortion Strain Energy
In terms of principle stresses2 2 2
0 1 2 3 1 2 2 3 3 1
1( ) 2 ( )
2U
E
2
0 1 2
12 (1 )
2U I I
E In terms of (total) stress invariants
2
3 3(1 2 )b
EK
221
0 1 2
1( 3 )
18 6b
IU I I
K
2 2 2 2
0 1 2 1 2 3 1 2 2 3 3 1
1 1( 3 )
6 6distortionU I I
Bulk modulus,
Slide 29
2 2 2
1 2 2 3 3 1
1( ) ( ) ( )
12
or
Example: Uniaxial tension
1 0 2 3; 0 2
0 0
12
12distortionU
2 2 2
0 1 2 2 3 3 1 2
1( ) ( ) ( ) 3
2J
hydrostatic deviatoric
Ie. same as the von Mises criterion
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Hardness
For indentation, the original motivation was to
measure plastic properties through a
parameter called hardness
Hardness is the resistance to penetration by the
indenter that leaves permanent deformation.
0H C
For hardness defined as proj
P
A
It is found (Tabor) that
The constant C is the constraint factor and has a
value of:
• ~3 for high ratio materials eg. metals
• ~1.5 for low ratio materials eg. polymers0E
0E
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Hertz (spherical) inelastic contact
Maximum shear stress in the contact stress
field of two solids of revolution occurs
below the surface on the axis of symmetry
Solid lines:
Hertz contact
, ,r z Are principle stresses
on this axis
r On the z-axis, so critical principle stress
difference for von Mises stress equation is z r
1
2 2
0 0
1(1 ) 1 tan
2(1 )
r z a
p p a z z a
2 2
0
1
(1 )
z
p z a
0
4
(1 )
ap
R
z r Is max value of at 00.62 p 0.48a
0 02.8 1.60p k Von Mises
yield (Slide 70)
3 2 3 3 23 3
0 0
(1.6)
12 (1 ) 12 (1 )yield yield
R RP p
( 0.3)
where
and
Slide 56 along z-axis:
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Sharp Indentation ProbesGenerally nanoindentation is performed with sharp pyramidal indenters
a
d
Ratio of contact radius with
depth of penetration is constant
giving geometric similarity
Scale invariant deformation
with constant strain
Value of strain is about 8% for
Vickers and Berkovich tips
Singular geometry at the tip is important
practical matter
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Instrumented IndentationBulychev, Alekhin, Shorshorov (1970’s), Oliver, Hutchings and Pethica (1983)
Tabor, essential physics of unloading of elastic-plastic materials after
indentation:
1. Plastic deformation is not recovered: All is elastic
2. Under conical indentation, residual impression elastically recovers
depth, but diameter remains unchanged.
*2projAdP
Edh
h
P
rheh
th
A
B
C
tP
2 2
1 2
*
1 2
1 11
E E E
projA Projected area
of contact
Determine Aproj with a shape function for
the tip for some depth: eg. orrhth
rh was chosen based off of TEM
replication data as best choice
* 12
proj
dPE
dhA t
proj
PH
A
NB:
h d
It was shown that for any smooth body of revolution, the initial unload slope is given
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Elastic Flat Punch Unloading
Cylindrical flat punch elastic solution for rigid punch:
*2P aE h
2* * *2 2 2
projAdP aaE E E
dh
*2dP A
Edh
True for all axially symmetric
indenters (Oliver-Pharr)
t
e
PdP
dh hAlso from diagram
Slide 3821
2
v P
E ad
2
*
1 1
E E
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Sharp Punch: Doerner & Nix
Doener and Nix (1986) key observation:
Initial slope of unloading curve for Berkovich
indenter is linear
Due to combination of indenter shape and
deformation geometry
Key abstraction: Can treat as unloading of
cylindrical flat punch (initially)
*2projAdP
Edh
Stage 1
Stage 2
Stage 3
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Doerner Nix
a
h
r
ahrh
eh
r fph
e fph
ph
h
P
rh eh
p r fph h
a e fph h
th
A
B
DC
tP
a e fph h
Consider the imaginary flat punch of radius a
The linear unload slope implies unloading of
the flat punch to the surface and
p r fph h
224.5Berk
proj pA h
is value to be plugged into tip shape function.
Eg. * 12
24.5p
dPE
dhh
24.5
t
p
PH
h
Measure: Peak displacement ht, unload displacement hr
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Oliver and Pharr
Sneddon: Load on axi-symmetric indenter of revolution indented into elastic
half space:mP h
h Is elastic displacement in surface
,m Are constants based on materials and
geometry, Eg. m=1 flat cyclinder, m=2
cones, m=1.5 spheres/paraboloids
Unloading of elastic-plastic materials: Under
indentation residual impression recovers only
depth, not diameter
*2projAdP
Edh
h
P
rheh
p rph h
a eph h
th
A
B
DC
tP
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Oliver and Pharr
c sh h h
* 2proj
SE
A
( , , )t t tP h S
dPS
dh
Measure at the peak:
Characterize the projected contact area via
the tip shape function at the peak:
( )proj cA F h
At any point:
c t sh h h
So must determine the surface deflection at the contact perimeter..sh
where
To extract elastic modulus:
t
Problem of elastic contact
hf
ht-hf
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Oliver and Pharr
This depends on indenter geometry. For cone (Sneddon):
2( )s t fh h h
From shape of elastic surface just
outside of contact region
( )zu r a
The quantity is is the elastic component of the displacement( )t fh h
From the elastic load vs. displacement relation for a cone indenter (Sneddon)
( ) 2 tt f
t
Ph h
S
ts
t
Ph
S 2
( 2)
Conical indenter
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An atomic limit for nanoindentation
How do we extend deformation to the smallest of length scales?
Do the deformation mechanics scale?
What does an atomic scale indentation crater look like? Is it stable?
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FIM – Field Evaporation
Indenter (tip) engineering by field evaporation
W He
Field ion microscope
Imaging : Voltage 4.5 kV
Field Evaporation : Voltage 5.2 kV
Single
Atom Tip
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Uses of a field evaporation point probe
Low energy e- holography
Surface forcesSTM/AFM tip definition
Nanoindentation: geometric BCNanoforming: limits
Contacts in nanoelectronics
Additive: W, Na, Cr, Ni, etc.
Mesoscopic and atomic level control:
M ~ d / Rtip
X-ray flash radiography
Coherent LE e- sourceUltra-bright
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Size and symmetry limits on indentation
Carrasco et. al., Phys. Rev. B 68 (2003)
50 nm
What are the size limits to fabrication?
Minimum size Surprising shapes
Do the laws of plasticity work at the nanoscale?
Cu 001 surface
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Nanoindentation defect processes
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Nanoindentation defect processes
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More information
See: G.L.W. Cross, J. Phys. D., 39 p. R1-R24 (2006).
G. L. W. Cross, et. al., Nature Materials, 5p. 370-76 (2006).
Rowland, H. et. al., Science, 302 p. 720-23 (2008).