Measuring the constitutive behavior of viscoelastic

solids in the time and frequency domain using flat 

punch nanoindentationpunch nanoindentation

E. G. Herbert, W. C. Oliver, A. L. Lumsdaine

Agilent Technologies, Oak Ridge, TN

G. M. PharrUniversity of Tennessee, Knoxville;

Materials Science and Technology Division,Oak Ridge National Laboratory, Oak Ridge, TN

NANOINDENTATION:  A CLASSIC APPLICATION

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 totime‐dependent behaviorp– Flat punch indentation, complex test geometry– DMA, triple clamp fixture, complex test geometry– Uniaxial compression simple test geometryUniaxial 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

NANOINDENTATION & DMA COMPARISON

PDMS, Tg ~ 120oC

C. C. White et al., Mater. R S S P 841Res. Soc. Symp. Proc. 841(2005) 

MODELING THE INSTRUMENTATION

i d d i l i• Measure time‐dependent material properties

• Then we need to understand the time‐dependent 

properties of the measurement tool

MODELING THE INSTRUMENTATION

FREE SPACENano Indenter® XP:

ti &&&Raw displacement, ± 1 mm

KhhChmeF tio ++=ω

( )φω −= tioehth )(2cos ωφ m

hF

K oS +=

( )( ) 1212222

−

⎥⎦⎤

⎢⎣⎡ +−= CmK

Fh

o

o ωω

Cωφsino

hF

C =

ho

2tanω

ωφmK

C−

=ωoh

MODELING THE INSTRUMENTATION

1 degree off d Z

Nano Indenter® XP:FREE SPACE

freedom, Z

ti &&&Raw displacement, ± 1 mm

KhhChmeF tio ++=ω

( )φω −= tioehth )(2cos ωφ m

hF

K oS +=

( )( ) 1212222

−

⎥⎦⎤

⎢⎣⎡ +−= CmK

Fh

o

o ωω

Cωφsino

hF

C =

ho

2tanω

ωφmK

C−

=ωoh

MODELING THE INSTRUMENTATION

1 degree off d Z

Nano Indenter® XP:FREE SPACE

freedom, Z

ti &&&Raw displacement, ± 1 mm

KhhChmeF tio ++=ω

( )φω −= tioehth )(2cos ωφ m

hF

K oS +=

( )( ) 1212222

−

⎥⎦⎤

⎢⎣⎡ +−= CmK

Fh

o

o ωω

C

FUNCTION OFPOSITION

ωφsino

hF

C =

ho

2tanω

ωφmK

C−

=ωoh

Measured stiffness and dampingin free space position = 18 8 μm

0

200 1000in free space, position = 18.8 μm

N/m

) Dam

400

-200

0

600

800Measured stiffnessModel

cos φ

(

mping, Fφω cos2 os h

FmK =− FREE SPACE

-600

-400

400Measured dampingF o /

h o c

Fo / h

o sioh

φω sino

o

hF

C =

-1000

-800200

Measured dampingModel

ffnes

s, in φ

(N/

-12000

1 10

Stif /m

)

Frequency (Hz)Frequency (Hz)

Measured stiffness and dampingin free space position = 18 8 μm

0

200 1000in free space, position = 18.8 μm

N/m

) Dam

400

-200

0

600

800Measured stiffnessModel

cos φ

(

mping, Fφω cos2 os h

FmK =− FREE SPACE

-600

-400

400Measured dampingF o /

h o c

Fo / h

o sioh

φω sino

o

hF

C =

-1000

-800200

Measured dampingModel

ffnes

s, in φ

(N/

-12000

1 10

Stif /m

)

Frequency (Hz)

m = 12.15 gKs = 95.4 N/mC = 2.81 Ns/mFrequency (Hz) /

ADD THE CONTACT

COUPLED2cos ωφ m

hF

S o +=ho

ωφsin

o

o

hF

C =

COUPLED RESPONSE = SAMPLE + INSTRUMENT

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

space) (free inst.

2

coupled

2contact coscos ωφωφ m

hF

mhF

Ko

o

o

o

PHASOR DIAGRAM: PHYSICAL INSIGHT

FREE SPACEDamped, forced oscillator

, N/m

)ng

, Cω

,(d

ampi

φω sinoFC =

ary

axis

o

o

hF

φω sinoh

C =

φω cos2 os

FmK =−

mag

ina

Real axis (stiffness K mω2 N/m)

φ

φo

s h

Im Real axis (stiffness, Ks-mω , N/m)

PHASOR DIAGRAM: PHYSICAL INSIGHT

Damped, forced oscillator

, N/m

)

FREE SPACE

ng, C

ω,

(dam

pi

φω sinoFC = COUPLED

ary

axis

o

o

hF

φω sinoh

C = COUPLED

φω cos2 os

FmK =−

mag

ina

Real axis (stiffness K mω2 N/m)

φ

φo

s h

Im Real axis (stiffness, Ks-mω , N/m)

PHASOR DIAGRAM: PHYSICAL INSIGHT

FREE SPACEDamped, forced oscillator

ω, N

/m)

ng, C

eqω

o

o

hF

φω sinoFC = COUPLED(d

ampi

n

φω sino

eq hC =

φω cos2 oeq

FmK =−

COUPLED

ry a

xis

(

φo

eq h

mag

inar

Real axis (stiffness K mω2 N/m)

φ

Im Real axis (stiffness, Keq

-mω , N/m)

PHASOR DIAGRAM: PHYSICAL INSIGHT

Depends on sample properties and the

Damped, forced oscillator

ω, N

/m)

FREE SPACE

p p p pgeometry of the contact

ng, C

eqω

o

o

hF

COUPLEDφω sinoFC =(d

ampi

n

COUPLEDCOUPLEDφω sino

eq hC =

φω cos2 oeq

FmK =−ry

axi

s ( COUPLED

φo

eq h

mag

inar

Real axis (stiffness K mω2 N/m)

φ

Im Real axis (stiffness, Keq

-mω , N/m)

FROM S AND Cω→ E’ AND E”

Phasor diagram of a linearviscoelastic solid

ss, P

a)

22* EEE ′′+′=

Phasor diagram of experimental measurements

ω, N

/m)

SAMPLE RESPONSE

cous

str

es

oEσ*

EEE +=

EiEE ′′+′=*

mpi

ng, C

ω

o

hF

F

SAMPLE RESPONSE

y ax

is (v

is

o

oEε

=

φεσ sin

o

oE =′′

y ax

is (d

am ohφω sin

o

o

hFC =

Imag

inar

yR l i ( l ti t P )

φφ

εσ cos

o

oE =′

Imag

inar

y

R l i ( tiff S N/ )

φφcos

o

o

hFS =

Real axis (elastic stress, Pa)Real axis (stiffness, S, N/m)

The fundamental equation of nanoindentation:

1π S 1 ωπ C)1(12

2νβ

π−=′

ASE )1(1

22νω

βπ

−=′′A

CE

DMA VS. NANOINDENTATION

Highly plasticized polyvinylchloride,the complex modulus at 22 oC

0.91

DMANanoindentation

10DMANanoindentationM

Pa)

0.50.60.70.80.9

Fact

or (-

)

6789

odul

us (M punch diameter = 100 μm

0.3

0.4

Loss

F4

5

6

tora

ge M

o

0.21 10

Frequency (Hz)

31 10

St

Frequency (Hz) q y ( )q y ( )

NEXT STEPS

• Do a better job comparing flat punch experiments to tests with simpler geometrywith simpler geometry.

• Study creep compliance with a flat punch.

• Combine frequency specific results with creep compliance.

COMPRESSION SAMPLES

COMPRESSION & INDENTATION

910

i i l i

)(cos factorgeometryhF

Eo

o φ=′

)(i f ttF

E o′′

789 uniaxial compression

1 mm dia. flat punch100 μm dia. flat punch

a)

)(sin factorgeometryhF

Eo

o φ=′′

5

6E'

(MPa

Geometry factors:

3

4

E

Compression:AL

FREQUENCY DOMAIN3

0.01 0.1 1 10 100Indentation: )1(11

22υ

βπ

−A

Frequency (Hz)

1uniaxial compression

)(cos factorgeometryhF

Eo

o φ=′

)(i f ttF

E o′′ uniaxial compression1 mm dia. flat punch100 μm dia. flat punch

r (-)

)(sin factorgeometryhF

Eo

o φ=′′

0 1Fact

orGeometry factors: 0.1

Loss

Compression:AL

FREQUENCY DOMAIN

0.01 0.1 1 10 100Indentation: )1(11

22υ

βπ

−A

Frequency (Hz)

205

79.6

80m

N)

Displace

transient response

Indentation:

PH =ασ hαε

Compression:

P=σ LΔ

=ε

195

200

78 4

78.8

79.2

On

Sam

ple

(m

ement Into S6 mN step load

transient response

AH =ασ

Dαε

A=σ

L=ε

LPLAtD Δ

=)()1(

)(2)( 2ν−=

PtRhtJc

10-6

190

195

77.6

78

78.4

2300 2400 2500 2600 2700 28002900

Load

O

urface (μm)

J (t), flat punch indentation2 /N)

Time On Sample (s)

10-7 c( ), p

(diameter = 983 μm)

D(t), uniaxial compression

D(t)

(m2

o

ttDσε )()( =

8

TIME DOMAIN

10-8

10-3 10-2 10-1 100 101 102 103

Creep Time (s)

TRANSFORMING FROM FREQUENCY TO TIME

4 term Prony series:

EJ′

′

FREQUENCY DOMAIN:

22 EEJ

′′+′=′

∑+=′4

220iJJJ ∑

= +1220 1i i ωτ

TIME DOMAIN:

t

∑=

−

−+=4

10 )1()(

i

t

iieJjtD τ

)(cos factorgeometryhF

Eo

o φ=′4

6

0.05 Hz Oscillation1 mm Dia. Flat Punch ho

)(sin factorgeometryhF

Eo

o φ=′′

-2

0

2

Load

(mN

)

3x10-7

5x10-7

FREQUENCY DOMAIN-6

-4

-1800 -1200 -600 0 600 1200 1800

L

3x102 /N

)22 EE

EJ′′+′

′=′

1800 1200 600 0 600 1200 1800Displacement (nm)

2x10‐7

88x10-81x10-7

uniaxial compression

1 mm dia. mm flat punchJ' (m

2EE +

Geometry factors:

4x10-8

6x10-8Compression:

AL

0.01 0.1 1 10 100Frequency (Hz)

Indentation: )1(112

2υβ

π−

A

J’ (FREQ. DOMAIN) FIT TO PRONY SERIES

3.2x10-7fit parameters:

= 7 5198E 03

FLAT PUNCH INDENTATION

E ′′

FREQ. DOMAIN:

2.4x10-7

2.8x10-7

N)

fit parameters:J

0 = 3.6960E-08

J 8 2830E 08

τ1

= 7.5198E-03

τ2 = 3.8620E+00

τ3 = 4.1034E-02

τ4 = 2.9883E-01

22 EEEJ

′′+′=′

∑+=′4

iJJJ

1.6x10-7

2x10-7

J' (m

2 /N J1 = 8.2830E-08

J2 = 4.6922E-08

J3 = 8.8235E-08

J4 = 7.5425E-08

4 ∑= +

+=1

220 1i i

JJωτ

8x10-8

1.2x10-7

nanoindentation data4 term parametric modelc r e fit

J 4 TIME DOMAIN:

∑=

−

−+=4

10 )1()(

i

t

iieJjtD τ

4x10-8

10-2 10-1 100 101 102 103

curve fit

ω (rad/s)ω (rad/s)

MAXIMIZING THE TIME AND FREQUENCY RANGE

3x10-7

5x10-7

D(t), uniaxial compression,measured in the time Combining indentation 

3x10

2 /N)

measured in the timedomain

s 10H0101.0

=s002.01.0=

data acquired in the 

frequency and time 

domain allows the PVC 

8x10-81x10-7

D(t)

(m2 Hz01.0s002.0

Hz 50reference material to be 

characterized over nearly 

6 decades in time (2x10‐3

4x10-8

6x10-8

D Predicted from flat punchnanoindentation dataacquired in the frequencydomain, 0.01 < f < 50 Hz

to 6x102).

10-5 10-4 10-3 10-2 10-1 100 101 102 103

Creep Time (s)p ( )

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 pfrequency and time domains.  

* It is possible to combine frequency and time domain data from a flat f ’punch indentation experiment and therefore characterize the sample’s 

behavior over the widest possible range of time and frequency.

GEOMETRY OF THE CONTACT

Advantages:– Known contact area

Circular flat punch:

Known contact area– Area not affected by creep or 

thermal drift 

Disadvantages:Disadvantages:

– Full contact

– Stress concentration

Any tip geometry, consider:

– Steady‐state harmonic motion

– Linear viscoelasticityLinear viscoelasticity

• Compression distance

• Oscillation amplitude

3Pre-Compression Dependence

33 μm5 μm10 μm1M

Pa)

punch dia. = 103 µm

101

15 μm20 μm

ulus

(M

Loss FStorage

ho = 50 nm

0.6

0.8

e M

odu Factor

0.4AVG LF (-)AVG LF (-)AVG LF (-)AVG LF ( )St

orag

e (-)

Loss factor

1 0.21 10

AVG LF (-)S

F (H )Frequency (Hz)

3Amplitude Independence

3

50 nm100 nm500 nmM

Pa) punch dia. = 103 µm

101

500 nm1500 nm3000 nm

ulus

(M

Loss FStorage

comp. dist. = 3 µm

0.6

0.8

e M

odu Factor

p µ

0.4AVG LF (-)AVG LF (-)AVG LF (-)AVG LF (-)St

orag

e (-)

Loss factor

1 0.21 10

AVG LF (-)S

F (H )Frequency (Hz)

22 C, 1 mm dia. punch15 C, 100 μm dia. punch

100 10 C, 100 μm dia. punch5 C, 100 μm dia. punch

Pa)

E' (M

P

10

1 10Frequency (Hz)

22 C, 1 mm dia. punch15 C 100 μm dia punch

10015 C, 100 μm dia. punch10 C, 100 μm dia. punch5 C, 100 μm dia. punch

Pa)

Shift factors:

E' (M

P 5 to 22: 20010 to 22: 140

15 to 22: 8

10

1 10 100 1000 104 105

ω*At (rad/s)

1000loss factor

100

1

a)

Loss

loss factor

storage modulus

E' (M

Pa

s Factor 10

0 1

(-)Freq range: 1 to 10 kHz

0.110 100 1000 104 105

ω (At) (rad/s)

THANKS EVERYONE!