3. MOTIVATION
What we’re after:
– Constitutive behavior of small volumes of viscoelastic solids
subjected time varying excitation over as wide a range of time
or frequency as possible.
Extend the applicability of nanoindentation to
time‐dependent behavior
p
– Flat punch indentation, complex test geometry
– DMA, triple clamp fixture, complex test geometry
– Uniaxial compression simple test geometry
compression, simple test geometry
• Material’s response in the frequency domain ‐ short time
• Material’s response in the time domain ‐ long time
• Bringing them together
• Do it all with flat punch indentation
• Material selection: Highly plasticized PVC
6. MODELING THE INSTRUMENTATION
Nano Indenter® XP:
FREE SPACE
Raw displacement, ± 1 mm
&& &
Fo e iωt = mh + Ch + Kh
h(t ) = ho e i (ωt −φ )
F
K S = o cos φ + mω 2
ho
)
((
−1
)
ho ⎡ ⎤
12
2
= ⎢ K − mω 2 + ω 2C 2 ⎥
F sinφ Fo ⎣ ⎦
C= o
ho ω Cω
tan φ =
K − mω 2
7. MODELING THE INSTRUMENTATION
Nano Indenter® XP:
FREE SPACE
1 degree of
freedom, Z
fd Z
Raw displacement, ± 1 mm
&& &
Fo e iωt = mh + Ch + Kh
h(t ) = ho e i (ωt −φ )
F
K S = o cos φ + mω 2
ho
)
((
−1
)
ho ⎡ ⎤
12
2
= ⎢ K − mω 2 + ω 2C 2 ⎥
F sinφ Fo ⎣ ⎦
C= o
ho ω Cω
tan φ =
K − mω 2
8. MODELING THE INSTRUMENTATION
Nano Indenter® XP:
FREE SPACE
1 degree of
freedom, Z
fd Z
Raw displacement, ± 1 mm
&& &
Fo e iωt = mh + Ch + Kh
h(t ) = ho e i (ωt −φ )
F
K S = o cos φ + mω 2
ho FUNCTION OF
)
((
−1
)
ho ⎡ ⎤
12
2
= ⎢ K − mω 2 + ω 2C 2
POSITION ⎥
F sinφ Fo ⎣ ⎦
C= o
ho ω Cω
tan φ =
K − mω 2
9. Measured stiffness and damping
in free space position = 18 8 μm
space, 18.8
1000
200
ffness, Fo / ho cos φ (N/m)
Dam
mping, Fo / ho si φ (N/
0
800
Measured stiffness
Model
-200
Fo
K s − mω 2 = cos φ 600 FREE SPACE
F
c
ho
-400
400
Fo
Cω = sinφ
ho
400
-600
Measured damping
in
Model
-800
200
-1000
Stif
/m)
0
-1200
1 10
Frequency (Hz)
10. Measured stiffness and damping
in free space position = 18 8 μm
space, 18.8
1000
200
ffness, Fo / ho cos φ (N/m)
Dam
mping, Fo / ho si φ (N/
0
800
Measured stiffness
Model
-200
Fo
K s − mω 2 = cos φ 600 FREE SPACE
F
c
ho
-400
400
Fo
Cω = sinφ
ho
400
-600
Measured damping
in
Model
-800
200
-1000
Stif
/m)
m = 12.15 g
0
-1200
Ks = 95.4 N/m
1 10
C = 2.81 Ns/m
/
Frequency (Hz)
11. ADD THE CONTACT
COUPLED
Fo
cos φ + mω 2
S=
ho
Fo sinφ
C=
ho ω
COUPLED RESPONSE = SAMPLE + INSTRUMENT
⎡F ⎤ ⎡F ⎤
= ⎢ o cos φ + mω 2 ⎥ − ⎢ o cos φ + mω 2 ⎥
K contact
⎢ ho coupled ⎥
⎦ ⎢o inst. (free space) ⎥
h
⎣ ⎣ ⎦
12. PHASOR DIAGRAM: PHYSICAL INSIGHT
Damped, forced oscillator
magina axis (damping, Cω, N/m)
FREE SPACE
,
Fo
Cω = sinφ
ho
Fo
ho Fo
ary
K s − mω 2 = cos φ
ho
φ
Im
Real axis (stiffness, Ks-mω2, N/m)
(stiffness mω
13. PHASOR DIAGRAM: PHYSICAL INSIGHT
Damped, forced oscillator
magina axis (damping, Cω, N/m)
FREE SPACE
,
Fo
Cω = sinφ COUPLED
ho
Fo
ho Fo
ary
K s − mω 2 = cos φ
ho
φ
Im
Real axis (stiffness, Ks-mω2, N/m)
(stiffness mω
14. PHASOR DIAGRAM: PHYSICAL INSIGHT
maginar axis (dampin Ceqω, N/m)
Damped, forced oscillator
FREE SPACE
ω
ng,
Fo
ho
Fo
Ceq ω = sinφ COUPLED
(
ho
Fo
K eq − mω 2 = cos φ
ry
ho
φ
Real axis (stiffness Keq-mω2, N/m)
Im
(stiffness, mω
15. PHASOR DIAGRAM: PHYSICAL INSIGHT
maginar axis (dampin Ceqω, N/m)
Damped, forced oscillator
FREE SPACE
Depends on sample properties and the
p ppp
ω
geometry of the
contact
ng,
Fo
ho
Fo
Ceq ω = sinφ COUPLED
COUPLED
(
ho
Fo
K eq − mω 2 = cos φ
ry
ho
φ
Real axis (stiffness Keq-mω2, N/m)
Im
(stiffness, mω
16. FROM S AND Cω → E’ AND E”
Phasor diagram of Phasor diagram of a linear
Imaginary axis (viscous stres Pa)
mping, Cω N/m)
experimental measurements viscoelastic solid
ss,
E * = E ′ 2 + E ′′ 2
ω,
SAMPLE RESPONSE
SAMPLE RESPONSE
E * = E ′ + iE ′′
Fo
σo
Imaginary axis (dam
ho
E* =
F
Cω = o sinφ εo
ho σo
sinφ
E ′′ =
εo
y
y
σo
Fo
cos φ
S= E′ = cos φ
εo
ho
φ φ
Real i ( tiff
R l axis (stiffness, S N/ )
S, N/m) Real i ( l ti t
R l axis (elastic stress, Pa)
P)
The fundamental equation of nanoindentation:
π1S π 1 Cω
E′ = (1 − ν 2 ) E ′′ = (1 − ν 2 )
2β A 2β A
17. DMA VS. NANOINDENTATION
Highly plasticized polyvinylchloride,
the complex modulus at 22 oC
MPa)
DMA
DMA
1 Nanoindentation
Nanoindentation
10 0.9
odulus (M
punch diameter = 100 μm
Loss Factor (-)
9 0.8
0.7
8
0.6
7
0.5
torage Mo
F
6
0.4
5
0.3
4
St
0.2
3
1 10
1 10
Frequency ( )
q y (Hz)
Frequency (Hz)
q y( )
20. COMPRESSION & INDENTATION
Fo
E′ = cos φ (geometry factor )
ho
10
F
E ′′ = o sinφ (geometry f t ) uniaxial compression
i il i
i t factor 9
ho 1 mm dia. flat punch
8
100 μm dia. flat punch
7
a)
E' (MPa
6
5
Geometry factors:
E
4
L
Compression:
A
FREQUENCY DOMAIN
3
π1 1
(1 − υ 2 )
Indentation:
2β A
0.01 0.1 1 10 100
Frequency (Hz)
21. Fo
E′ = cos φ (geometry factor )
ho
1
F
E ′′ = o sinφ (geometry f t ) uniaxial compression
i t factor
ho 1 mm dia. flat punch
100 μm dia. flat punch
Loss Factor (-)
r
Geometry factors:
0.1
01
L
Compression:
A
FREQUENCY DOMAIN
π1 1
(1 − υ 2 )
Indentation:
2β A
0.01 0.1 1 10 100
Frequency (Hz)
22. Displace
Compression: Indentation:
80
205
ΔL
mN)
P P h
79.6
ε=
σ= σα H = εα
Load On Sample (m
ement Into Surface (μm)
transient response
L
A A D
79.2
200
A ΔL 2Rh(t )
J c (t ) =
D( t ) =
78.8
P (1 − ν 2 )
6 mN step load
LP
O
78.4
78 4
195
10-6
78
77.6
190
2300 2400 2500 2600 2700 2800 2900
Time On Sample (s)
D(t) (m2/N) J (t), flat punch indentation
( ), p
c
-7
(diameter = 983 μm)
10
ε (t )
D( t ) = D(t), uniaxial compression
σo
TIME DOMAIN
10-8
8
10-3 10-2 10-1 100 101 102 103
Creep Time (s)
23. TRANSFORMING FROM FREQUENCY TO TIME
4 term Prony series:
FREQUENCY DOMAIN:
E′
J′ =
E ′2 + E ′′2
4
Ji
J ′ = J0 + ∑
1+ τ i ω 2
2
i =1
TIME DOMAIN:
−t
4
D(t ) = j 0 + ∑ J i (1 − e τ i )
i =1
24. 6
Fo
0.05 Hz Oscillation
E′ = cos φ (geometry factor )
4 1 mm Dia. Flat Punch
ho
Load (mN)
2
Fo
E ′′ = sinφ (geometry factor )
0
ho
-2
L
-7
5x10
-4
FREQUENCY DOMAIN
-6
-7
1200 18003x10
-1800 -1200 -600
1800 1200 600 0 600
Displacement (nm)
2x10‐7
E′
J' (m2/N)
J′ = 2
E ′ + E ′′ 2
-7
1x10 uniaxial compression
Geometry factors:
8x10-8 1 mm dia. mm flat punch
-8
8
L 6x10
Compression:
A
-8
4x10
π1 1
(1 − υ 2 )
Indentation: 0.01 0.1 1 10 100
2β A
Frequency (Hz)
25. J’ (FREQ. DOMAIN) FIT TO PRONY SERIES
FLAT PUNCH INDENTATION
FREQ. DOMAIN:
-7
3.2x10 fit parameters:
E′
τ = 7.5198E-03
J′ =
7 5198E 03
1
E ′2 + E ′′2
2.8x10-7 τ2 = 3.8620E+00
τ3 = 4.1034E-02
fit parameters:
-7
2.4x10 4
J = 3.6960E-08
Ji
J ′ = J0 + ∑
τ4 = 2.9883E-01
0
N)
J' (m2/N
J = 8 2830E 08
8.2830E-08
1+ τ i ω 2
2
-7 1
i =1
2x10 J = 4.6922E-08
2
J = 8.8235E-08
1.6x10-7 3
J = 7.5425E-08
J
4
TIME DOMAIN:
-7
1.2x10
−t
4
D(t ) = j 0 + ∑ J i (1 − e τ i )
-8 nanoindentation data
8x10 4 term parametric model
i =1
curve
c r e fit
-8
4x10
10-2 10-1 100 101 102 103
ω (rad/s)
26. MAXIMIZING THE TIME AND FREQUENCY RANGE
5x10-7
D(t), uniaxial compression,
Combining indentation
-7 measured in the time
3x10 domain data acquired in the
frequency and time
D(t) (m2/N)
0.1
0 .1 domain allows the PVC
= 10 s
= 0.002 s
0.01 H
Hz
50 Hz
reference material to be
-7 characterized over nearly
1x10
8x10-8 6 decades in time (2x10‐3
D
Predicted from flat punch
to 6x102).
-8 nanoindentation data
6x10
acquired in the frequency
domain, 0.01 < f < 50 Hz
4x10-8
-5 -4 -3 -2 -1 0 1 2 3
10 10 10 10 10 10 10 10 10
Creep Time (s)
p ()
27. CONCLUSIONS
* Dynamic nanoindentation of viscoelastic solids requires robust
dynamic characterization of the measurement tool itself, a known
contact geometry, steady‐state harmonic motion, and linear
viscoelasticity.
* In the frequency domain Sneddon’s stiffness equation works
In the frequency domain, Sneddon s stiffness equation works
remarkably well.
* The Prony series model provides a valid path to transition between the
y p p
frequency and time domains.
* It is possible to combine frequency and time domain data from a flat
punch indentation experiment and therefore characterize the sample’s
f ’
behavior over the widest possible range of time and frequency.
28. GEOMETRY OF THE CONTACT
Circular flat punch: Advantages:
– Known contact area
Known contact area
– Area not affected by creep or
thermal drift
Disadvantages:
– Full contact
– Stress concentration
Any tip geometry, consider:
– Steady‐state harmonic motion
– Linear viscoelasticity
Linear viscoelasticity
• Compression distance
• Oscillation amplitude
29. Pre-Compression Dependence
3
3 μm
MPa)
punch dia. = 103 µm
5 μm
10 μm
ulus (M
1 μm
15
Loss F
Storage
20 μm
10
1
ho = 50 nm
Storage Modu
Factor (-)
0.8
0.6
e
0.4
AVG LF (-)
AVG LF (-)
Loss factor AVG LF (-)
S
AVG LF (-)
()
0.2
1
1 10
Frequency (Hz)
F (H )
30. Amplitude Independence
3
MPa)
punch dia. = 103 µm
50 nm
100 nm
ulus (M
500 nm
Loss F
1500 nm
10 3000 nm Storage
1
comp. dist. = 3 µm
p µ
Storage Modu
Factor (-)
0.8
0.6
e
0.4
AVG LF (-)
AVG LF (-)
Loss factor AVG LF (-)
AVG LF (-)
S
AVG LF (-)
0.2
1
1 10
Frequency (Hz)
F (H )
