MEASUREMENTS OF VISCOELASTIC PROPERTIES
BY NANOINDENTATION
By
GANG HUANG
Bachelor of Science in Mechanical Engineering
Huazhong University of Science and Technology
Wuhan, China
1997
Master of Science in Mechanical Engineering
Huazhong University of Science and Technology
Wuhan, China
2000
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
DOCTOR OF PHILOSOPHY
May, 2007
ii
MEASUREMENTS OF VISCOELASTIC PROPERTIES
BY NANOINDENTATION
Dissertation Approved:
Dr. Hongbing Lu
Dissertation Adviser
Dr. Don. A. Lucca
Dr. Andrew S. Arena
Dr. Leticia Barchini
Dr. Demir Coker
Dr. A. Gordon Emslie
Dean of the Graduate College
iii
ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor, Dr. Hongbing Lu, for his
supervision throughout my entire Ph.D. study. I appreciate his intelligent, perseverant,
warmhearted, and inspiring guidance. I am also grateful for his generous financial
support to allow me to complete my research work in the program. I would like to thank
all other members of my thesis committee, Dr. Don A. Lucca, Dr. Andrew S. Arena, Jr.
and Dr. D. Coker from the School of Mechanical and Aerospace Engineering, Dr. Leticia
Barchini from the Department of Mathematics. My thanks also extend to the faculty and
staff of the Mechanical and Aerospace Engineering. I would express respectful thanks to
Dr. Bo Wang for his guidance and advice in my research work. I would also appreciate
Dr. Rong Z. Gan from the School of Aerospace and Mechanical Engineering at the
University of Oklahoma for her diligent supervision during my research work on
nanoindentation of biological tissues.
I would also thank Dr. Jin Ma, Mr. Nitin Daphalapurkar, Ms. Haowen Yu, Dr.
Huiyang Luo for their constructive suggestions and discussions during my research work.
I would thank all other members in Dr. Hongbing Lu’s research group for the compatible
and enjoyable environment.
My research work was initially supported by the National Science Foundation under
CMS9985060, and was supported by NASA under grant NNL04AA4ZG with Dr.
Thomas S. Gates, the technical manager, as the technical director since early 2005.
iv
The work on nanoindentation of eardrum is supported by NIH (NIDCD R01DC006632).
I would express my sincere gratitude to my parents, my sister and my brother for their
unconditional support, understanding, and confidence on me all the time. Without their
endless and consistent support, none of my research work could be possible. This
dissertation is dedicated to my whole family.
v
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION........................................................................................................... 1
II. FUNDAMENTAL THEORIES ON NANOINDENTATION .................................... 10
2.1 Elastic indentation problem .............................................................................. 10
2.2 Local stressstrain analysis of indentation problem.......................................... 14
2.3 Linear viscoelasticity ........................................................................................ 18
2.4 Linearly viscoelastic indentation problem........................................................ 19
III. MEASUREMENTS OF CREEP COMPLIANCE (TIMEDOMAIN) ...................... 21
USING NANOINDENTATION ................................................................................ 21
3.1 Analytical prerequisites .................................................................................... 21
3.2 A corrected method to obtain creep function under a step loading .................. 24
3.3 Experiments ...................................................................................................... 27
3.4 Result and discussion........................................................................................ 28
IV. MEASUREMENTS OF COMPLEX CREEP COMPLIANCE (FREQUENCY
DOMAIN) USING NANOINDENTATION.............................................................. 32
4.1 Introduction....................................................................................................... 32
4.2 Theoretical background .................................................................................... 35
4.3 Experiments ...................................................................................................... 41
4.3.1 DMA experiments..................................................................................... 41
4.3.2 Nanoindentation experiments ................................................................... 42
4.4 Results and discussions..................................................................................... 44
4.4.1 DMA results.............................................................................................. 44
4.4.2 Nanoindentation results ............................................................................ 45
4.5 Conclusions....................................................................................................... 56
V. MEASUREMENTS OF TWO INDEPENDENT VISCOELASTIC .......................... 58
FUNCTIONS BY NANOINDENTATION................................................................. 58
5.1 Introduction....................................................................................................... 58
5.2 Analytical Background ..................................................................................... 59
5.2.1 Indentation by rigid axisymmetric indenters of arbitrary shape ............... 60
5.2.2 Berkovich indenter.................................................................................... 63
5.2.3 Viscoelastic solutions................................................................................ 66
5.3 Nanoindentation measurements ........................................................................ 70
5.4 Results and discussions..................................................................................... 72
5.5 Conclusions....................................................................................................... 80
VI. APPLICATION OF NANOINDENTATION TO MEASURING VISCOELASTIC
FUNCTION FOR THIN FILMS I: SINGLE WALL CARBON NANOTUBE
(SWNT)....................................................................................................................... 82
6.1 Introduction....................................................................................................... 82
vi
6.2 Analytical prerequisites .................................................................................... 84
6.3 Experimental details.......................................................................................... 86
6.3.1 SWNT/ polyelectrolyte films preparation................................................. 86
6.3.2 Measurements ........................................................................................... 88
6.4 Results and Discussions.................................................................................... 90
6.5 Conclusions....................................................................................................... 96
VII. APPLICATION OF NANOINDENTATION TO MEASURING VISCOELASTIC
FUNCTION FOR THIN FILMS II: TYMPANIC MEMBRANE ............................ 98
7.1 Introduction....................................................................................................... 98
7.2 Analytical prerequisites .................................................................................. 100
7.2.1 Measurements of the throughthickness properties using nanoindentation..
................................................................................................................. 100
7.2.2 Analytical solution for the measurement of inplane viscoelastic properties
of TM under a central concentrated load ................................................ 100
7.3 Nanoindentation experiments ......................................................................... 103
7.4 Results and discussions................................................................................... 107
7.4.1 Relaxation modulus measured from throughthickness nanoindentation
tests ......................................................................................................... 107
7.4.2 Analytical results of the Young’s relaxation modulus from inplane
nanoindentation tests on TM................................................................... 118
7.5 Conclusions..................................................................................................... 121
VIII. MEASUREMENTS OF RELAXATION MODULUS USING
NANOINDENTATION......................................................................................... 123
8.1. Analytical Prerequisite................................................................................... 124
8.2. Nanoindentation Measurements..................................................................... 129
8.3. Results and Discussions................................................................................. 132
8.4. Conclusions.................................................................................................... 142
IX. SUMMARY.............................................................................................................. 144
REFERENCES ............................................................................................................... 149
vii
LIST OF TABLES
Table Page
61 Comparison of modulus data for the SWNT/polyelectrolyte and polyelectrolyte films
determined from different approaches ........................................................................94
71 Results of modulus (steady state) from nanoindentation measurements...................121
viii
LIST OF FIGURES
Figure Page
11 A schematic for the mechanism of nanoindentation .....................................................2
12 A schematic indentation using conical indenter (Oliver and Pharr, 1992)....................2
13 A typical loaddisplacement curve of nanoindentation .................................................3
21 Indentation of elastic layer by cylindrical indenter (Ling, Lai and Lucca, 2002) .......13
22 Elastic indentation problem.........................................................................................15
31 A ramp loading history................................................................................................25
32 Constant loading rate history and step loading history................................................28
33 Loaddisplacement curves for Berkovich indenter and spherical indenter .................29
34 Experiment curve and fitting curve for loaddisplacement .........................................30
35 Creep compliance measured from nanoindentation ....................................................31
41 Comparison of storage compliance at 75 Hz computed by two methods (a) PC.
(b) PMMA............................................................................................................34
42 Geometry of the spherical indenter. (a) Schematic diagram of the indenter. (b) A
TEM image of the spherical indenter tip. ............................................................36
45 Nanoindentation output from oscillation on a constant rate loading at 10 Hz. (a)
Carrier loaddepth curve. (b) History of amplitude of harmonic load. (c)
Response of harmonic displacement amplitude. (d) Outofphase angle with
correction .............................................................................................................46
46 Comparison of contact radius results at 75 Hz using the LeeRadok approach
and the Ting approach. (a) Contact radius computed for PMMA under a
harmonic load superimposed on a step loading. (b) Contact radius computed for
PC under a harmonic load superimposed on a step loading. ..............................47
47 Nanoindentation output from oscillation on a step loading at 75 Hz. (a) Carrier
loaddepth curve. (b) The history of harmonic load amplitude. (c) Response of
harmonic displacement amplitude. (d) Outofphase angle with correction........50
ix
48 Complex compliance in shear from nanoindentation under a harmonic load
superimposed on a constant rate loading. ............................................................53
49 Complex compliance in shear from nanoindentation under a harmonic load
superimposed on a step loading. .........................................................................54
51 A schematic of indentation on a half space by an axisymmetric indenter ..................61
52 Berkovich, conical and spherical indenters .................................................................63
53 FEM modeling.............................................................................................................65
54 Nanoindentation loaddisplacement curves for PVAc ................................................72
55 Minimization results of loaddisplacement curves for PVAc .....................................74
56 Results of K(t) and μ (t) for PVAc from nanoindentation........................................74
57 Nanoindentation loaddisplacement curves for PMMA..............................................76
58 Minimization results of loaddisplacement curves for PMMA...................................77
59 Results of K(t) and μ (t) for PMMA from nanoindentation .....................................79
61 SEM and AFM images of SWNT/polyelectrolyte films. (a) SEM image of the cross
section of a 100layer SWNT/polyelectrolyte film; (b) AFM image of top surface of
a monolayer SWNT/polyelectrolyte film....................................................................87
62 Stressstrain curves of SWNT/polyelectrolyte and resin.............................................90
63 Experimental and fitted nanoindentation loaddisplacement curves. (a) Inpane
nanoindentation loaddisplacement curves for SWNT/polyelectrolyte films from
nanoindentation measurements and the fitting method; (b) Throughthickness
nanoindentation loaddisplacement curves for SWNT/polyelectrolyte films from
experiments and fitting method; (c) Nanoindentation loaddisplacement curves of
polyelectrolyte from experiments and fitting method................................................91
64 Uniaxial relaxation modulus of SWNT/polyelectrolyte film and polyelectrolyte film
measured by nanoindentation. (a) Inplane modulus measured by both fitting and
differentiation methods for SWNT/polyelectrolyte films; (b) Outof plane modulus
measured by both fitting and differentiation methods for SWNT/polyelectrolyte films;
(c) Modulus measured by both fitting and differentiation methods for
polyelectrolyte films. ................................................................................................93
71 A schematic of a thin plate cconstant rateed at the perimeter and loaded with
concentrated force at the center. ..............................................................................101
x
72 Image of the right TM (medial view). .......................................................................104
73 Schematic of throughthickness and inplane nanoindentation tests setup (a)
throughthickness test; (b) Inplane test...................................................................106
74 Throughthickness tests results for dry samples. (a) Loaddisplacement curves;
(b) relaxation modulus E(t). .....................................................................................108
75 Throughthickness tests results for wet samples. (a) Loaddisplacement curves for
the Posterior side of TM ; (b) Loaddisplacement curves for the Anterior side of TM;
(c) The measured E(t). .............................................................................................110
76 Results for dry TMs from inplane nanoindentation tests. (a) Loaddisplacement
curves; (b) Correlation of loaddisplacement curves between finite element analysis
and experiments; (c) E(t) determined from the analytical solution and finite element
method..................................................................................................................... 114
77 Loaddisplacement curves of wet TM from inplane indentation tests. (a) Sample
from posterior side of TM; (b) Sample form anterior side of TM. ...........................115
78 Results for wet posterior and anterior samples from inplane nanoindentation tests. (a)
Correlation of loaddisplacement curves between finite element analysis and
experiment; (b) E(t) determined from finite element method and the analytical
solution…...................................................................................................………...116
8 1 Schematic of conical indentation and the geometries of the Berkovich, conical
and spherical indenters............................................................................................. 125
8 2 Displacement history for nanoindentation tests of PMMA, PC and PU using
Berkovich indenter tip............................................................................................. 131
8 3 Nanoindentation loaddisplacement curves for PMMA, PC and PU using Berkovich
indenter tip .............................................................................................................. 132
8 4 Fitted and measured curves for PMMA using Eq. (813) (Berkovich indenter tip) 134
8 5 Results of E(t) for PMMA measured from different methods............................... 134
8 6 Fitted and measured curves for PC using Eq. (813) (Berkovich indenter tip) ....... 135
8 7 Young’s relaxation function E(t) for PC measured from different methods........... 136
8 8 Fitted and measured curves for PU using Eq. (813) (Berkovich indenter tip)....... 137
8 9 Young’s relaxation modulus E(t) for PU measured from different methods ......... 138
xi
8 10 Nanoindentation loaddisplacement curves for PMMA, PC and PU measured from
nanoindentation tests using a spherical indenter tip, plotted with the fitted curve as
described by Eq. (817)......................................................................................... 141
1
CHAPTER I.
INTRODUCTION
Nanoindentation technique for measurements of mechanical properties has been
developed since early 1980’s. The areas of its applications have been growing in the past
15 years due to the commercial availability nanoindentation instrumentation and the ease
of applying the technique to measure mechanical properties of very small amounts of
materials, such as thin solid films, wires, components in MEMS and NEMS, for which it
is a challenge to determine properties using conventional testing methods, such as tensile
or tortional tests.
Nanoindentation is an extension of conventional indentation technique to micron and
submicron scales. The conventional indentation tests have been established as a standard
method to measure mechanical properties of materials for more than one century (Dieter,
1986). Fig. 11 shows a schematic setup for nanoindentation. In a typical nanoindentation
test, or depth sensing test, a nanoidenter tip of certain geometry (usually, Berkovich,
spherical, conical, or flat punch) indents into the workpiece, while the load applied and
induced depth are recorded. The early understanding of indentation problem was
attributed to Hertz (1896), hunter (1960), Boussinesq (1885) and Sneddon (1965). Hertz
(1896) solved the problem of elastic contact between two spheres. The Hertzian problem
was investigated by Hunter (1960) for viscoelastic materials. Boussineq (1885) derived
2
the solution for stress and strain distribution of concentrated force on half space of elastic
materials. Sneddon (1965) generalized the relationship between load and depth of
indentation for elastic axisymmetric indentation problems. These solutions laid the
foundation for extracting mechanical properties (including viscoelastic properties) from
nanoindentation data.
Fig. 11 A schematic for the mechanism of nanoindentation
Fig. 12 A schematic indentation using conical indenter (Oliver and Pharr, 1992)
3
A contact profile from nanoindentation using a conical indenter is depicted in Fig. 12. In
this figure, h is the indentation depth (the tip displacement) that is composed of hs, the
depth at perimeter of the free surface, and hc, the contact depth, hf is the depth of
impression after load is fully removed and a is the contact radius. Fig. 13 shows a typical
form of nanoindentation load displacement data. Conventional indentation tests deal
primarily with relatively large deformation. The hardness of a material is obtained by the
maximum indentation force divided by the projected area of final impression measured
optically after the load is removed (Pethica, Hutchings, Oliver 1983).
Fig.13
Fig. 13 A typical loaddisplacement curve of nanoindentation
With the increasing applications of very small structures, such as MEMES (Mirco
ElectronicMechanical System), and very small amounts of material, such as thin film
deposited on substrates, indentation technique was developed to provide an approach for
4
measuring properties at micron and submicron scales. Oliver, Hucthings and Pethica
(1986) proposed a method to determine the hardness by calculating the contact area with
the depth of impression obtained after indentation load is removed. Doerner and Nix
(1991), Oliver and Pharr (1991, 1992) later refined the method to determine the contact
area through indenter shape calibration. Using the contact area at maximum load and by
considering Sneddon’s solution for an elastic indentation problem, they developed
methods to measure elastic modulus and hardness. In their improved methods, the contact
area is measured at submicron resolution without the necessity to image optically the
indent impression, as conventional indention tests did. Thus their improved methods have
simplified nanoindentation testing procedures significantly on elasticplastic materials.
While nanoindentation technique has been widely applied for elastic materials, it has
attracted increasing attention for measuring mechanical properties of viscoelastic
materials, such as polymers. Since the invention of synthetic polymers last century, they
have been increasingly used in mechanical and chemical engineering. Due to their high
specific strength, ease of fabrication and high corrosion resistance, they are anticipated to
play an important role in some engineering areas, such as medical, automotive and
aerospace industry. In these applications, how to measure reliably and effectively
mechanical properties of polymers is critical to understand the deformation and failure of
polymers and their components. In a variety of applications involving use of small
amounts of polymers, such as filaments and fibers used in medical implants, polymer
films deposited on substrates, the conventional methods are not suitable to measure some
mechanical properties, such as Young’s modulus in the throughthickness direction.
Consequently, nanoindentation becomes an important technique in these situations.
5
Despite the fact that nanoindentation technique for measurements of some properties,
such as Young’s modulus and hardness for elasticplastic materials, has been well
established and used widely, nanoindentation on viscoelastic materials is not fully
understood even in the regime of linear viscoelasticity. For nanoindentation on
viscoelastic materials in the regime of linear viscoelasticity, the viscoelastic properties in
both time domain and frequency domain are often of interest. In this dissertation,
methods are presented to measure viscoelastic functions in both time and frequency
domains for timedependent materials using nanoindentation.
Solutions to linearly viscoelastic contact mechanics problems have been derived in
the past. Radok (1960) found analytical solution to the viscoelastic contact problem
involving a spherical indenter; some results have been reported on indentation on
viscoelastic materials. Shimizu, Yanagimoto and Sakai (1999), Sakai and Shimizu (2001)
investigated the viscoelastic response of soda lime silica glass through pyramidal
indentation at glass transition temperature. Cheng et al. (2000) measured the relaxation
modulus using a flatended cylindrical punch for a polymer that can be described by a
threeelement model. Vanlandingham et al. (2005) measured relaxation modulus under a
nearstep displacement and creep compliance using a nearstep loading and investigated
the applicability of linear viscoelasticity in nanoindentation on polymers. Cheng and
Cheng (2005) derived an expression of unloading stiffness for a linearly viscoelastic solid
under nanoindentation. These investigations are useful under their respective situations
but are far from complete.
For measurement of viscoelastic properties in frequency domain using
nanoindentation, Loubet et al. (1995) proposed a method based on analogy between
6
indention under cyclic loading and uniaxial counterpart, but there was no rigorous theory
to support it. To use nanoindentation to find the viscoelastic properties of a general
viscoelastic material in linear regime, consensus methods need to be developed, which is
the objective of this work. In this dissertation (Chapter 4), from the original definition, an
analytical method is derived to compute the complex creep compliance. And with the
data provided by dynamic indentation experiments, the creep compliance for some solid
polymer materials is obtained. They are compared with conventional counterparts to
validate the method. The details will be presented later in this study.
Poisson’s ratio, as one of two independent mechanical properties for isotropic linearly
elastic (or viscoelastic) materials, plays a very important role in the deformation of
materials, and is often assumed as constant in the measurement of some material
properties, such as creep compliance in nanoindentation. While it is adequate to assume
the Poisson’s ratio as a known constant value for viscoelastic materials, such as
polymers, well below the glass transition temperature, analysis based on assumption of a
constant Poisson’s ratio above glass transition temperature can induce considerable
errors, since the Poisson’s ratio of the viscoelastic material under this situation varies
with time. In this dissertation, a method is developed to measure two independent
viscoelastic functions using nanoindentation without recourse to the assumption of a
constant Poisson’s ratio.
This dissertation consists of nine chapters. In the first three chapters, background on
the measurements of viscoelastic properties using nanoindentation is reviewed and
methods for measuring linearly viscoelastic functions are developed and validated. In
7
the two chapters that follow, the developed methods are applied to two different films to
determine the relaxation modulus. The dissertation is organized as follows.
In Chapter 2, previous methods from literature for measurements of elastic properties
are reviewed; some theories on elastic and contact mechanics problems are summarized.
In Chapter 3, methods are developed for measuring creep compliance in time domain
using nanoindentation under two quasistatic loading histories, namely, a constant rate
loading and a step loading. Equations for determining creep compliance are derived and
validated using two solid polymers, polymethyl methacrylate (PMMA) and
polycarbonate (PC).
In chapter 4, a method is developed for measuring creep compliance in frequency
domain using nanoindentation under dynamic loading histories which are achieved by
superimposing a small oscillation upon a constant rate loading or step loading. Equations
for determining complex creep compliance are derived and validated using PMMA and
PC through comparing the nanoindentation data to data from Dynamic Mechanical
Analysis.
In Chapter 5, a method is developed to measure two shear relaxation modulus and
shear bulk modulus using nanoindentation assuming Poisson’s ratio as functions of time.
The method is validated on two bulk polymers, namely, poly(vinyl acetate) (PVAc) and
PMMA above and below the glass transition temperatures, respectively.
In Chapter 6, methods for measuring inplane and outofplane relaxation moduli are
applied to singlewall carbon nanotube composite films made by layer by layer assembly.
8
In Chapter 7, the outofplane relaxation modulus of Tympanic Membrane is
determined using the methods developed for measuring viscoelastic functions in time
domain; the inplane relaxation modulus is determined by a method based on the
correlation between loaddisplacement results from finite element analysis and those
from nanoindentation, and through the analytical analysis of the loaddisplacement data
for a clamped circular film subjected to a central concentrated force.
In Chapter 8, methods are presented to measure uniaxial/shear relaxation modulus for
linearly viscoelastic materials using nanoindentation. A constant rate displacement
loading history is applied in nanoindentation tests. Based on viscoelastic contact
analysis, uniaxial/shear relaxation modulus is extracted from nanoindentation loaddisplacement
data from nanoindentation experiments. The methods for direct
measurement of shear relaxation modulus can avoid the solution of an illposed problem
in conversion from creep functions determined using load control in nanoindentation.
Chapter 9 gives a thorough summary for the dissertation.
In measurements of mechanical properties of viscoelastic materials, one of interesting
and hardlyunderstood phenomena in nanoindentation is the appearance of negative
slope in the unloading loaddisplacement curve for some polymers. To date, in literature
there has been no quantitative explanation for this phenomenon. A rigorous approach to
address the negative phenomenon is suggested for implementation in future work.
Another work is to determine the master curves of viscoelastic properties for polymers
using nanoindentation. Determination of the master curve for polymers is very important
to predict the longterm behavior. In contrast from the measurements of viscoelastic
properties at room temperature at which Poisson’s ratio can normally be assumed as a
9
constant for polymers with much higher Tg than room temperature, the Poisson’ ratio
can no longer be assumed to be constant since in the development of master curves
nanoindentation has to be conducted at a series of elevated temperatures at which
Poisson’s ratio changes with time. To analyze data at elevated temperatures the methods
introduced in Chapter 5 for the measurements two independent viscoelastic functions
can be applied, and shear relaxation modulus (or bulk modulus) functions can be
measured at different temperatures to form master curves using the timetemperature
superposition principle.
10
CHAPTER II.
FUNDAMENTAL THEORIES ON NANOINDENTATION
Some theories on elasticity, linear viscoelasticity, that form the foundation for
nanoindentation technique, are summarized in this chapter. Linear elastic contact
mechanics will be introduced, followed by linear viscoelastic analysis.
2.1 Elastic indentation problem
For an elastic problem of contact between a spherical indenter and a halfspace,
Hertz (1896) derived the pressure (the normal stress) in the contact region
2 2
0 (1 )
4
r r
R v
p −
−
− =
μ
, 0 0 < r < r (21)
where μ is shear modulus, v the Poisson’s ratio, R the radius of the spherical indenter,
and r0 contact radius.
Integrating over the contact area yields the force applied by the indenter.
r r r dr
R v
P
r
−
−
=
0
0
2 2
0 2
(1 )
4
μ
, (22)
which leads to:
3
0 3(1 )
8
r
v R
P
−
=
μ
. (23)
11
According to Hunter (1960), for a rigid spherical indenter, if h«R
r = Rh 2
0 , (24)
where h is the displacement of tip of indenter.
From Eqs. (23) and (24), the relationship between loading and displacement for
spherical indentation is:
2
3
3(1 )
8
h
v
R
P
−
=
μ
. (25)
Sneddon (1965) derived the general expressions for depth and loading in terms of
indenter shape function. He also obtained relationship between loading and depth for
several indenter tips. His equations for flat and conical indenters are widely used. For
indentation by a flatend circular indenter tip, the Sneddon’s equation for loaddisplacement
is:
h
v
R
P
−
=
1
4μ
, (26)
where R is the radius of flat indenter.
For conical indenter, the equation is
2
(1 )
4 cot
h
v
P
−
=
μ
(27)
where is denoted as shown in Fig. 12.
It is noted that above equations are limited to rigid indenter, with an elastic modulus
assumed to be infinity. For the problem of a deformable indenter indenting into a halfspace,
the reduced modulus was introduced (FicherCripps, 1999). For conical indention,
the loadingdepth equation is then
12
8 cot 2
h
E
P r
= , (28)
where Er
is reduced modulus, which is expressed as
i
i
r E
v
E
v
E
1 1 2 1− 2
+
−
= , (29)
where v and vi are the Poisson’s ratios of the sample material and the indenter,
respectively, and E, Ei are the Young’s modulus data of the sample material and the
indenter, respectively.
A detailed derivation for solutions to indentation problem of elastic half space by
rigid cylindrical, conical and spherical indenters, as well by elastic indenters was given
by Ling, Lai and Lucca (2002). They also gave solutions to indentations of rigid indenters
into an elastic layer supported by a rigid plane. For example, for indentation by a
cylindrical indenter into a elastic layer with a height of h supported by a rigid substrate
(Fig.21), Eq. (26) is changed into
μ
d
Ha
P v
( )
4
(1 ) 1
0 1 =
−
, (210)
where
= − [ + + − ]
1
1 0 1 ( ) ( ) ( )
1
( ) 1
K K d , (211)
and K is determined by
[ ]
d
h
a u
v v
v e v
h
a
u K
+ − + −
− − + + −
= −
cos
4(1 ) (3 4 ) sinh
(3 4 )(sinh ) (1 ) 4(1 )
( )
0 2 2 2
2
. (212)
This result is very conducive for the investigation of the mechanical behavior of solid
thin film using nanoindentation.
13
Fig. 21 Indentation of elastic layer by cylindrical indenter (Ling, Lai and Lucca,
2002)
Usually the indenters used are made of diamond so that they can be considered as
rigid materials, especially for indentation on polymers as the difference in modulus data
is two orders of magnitude. Consequently, in this dissertation, the equations for elastic
indentation can be applied readily without considering deformation of the indenter.
In indentation on elasticplastic materials, the contact area between indenter and
sample material is needed to extract properties, such as elastic modulus and some other
properties. Doerner and Nix (1986) found a linear relationship between contact stiffness
and square root of projected contact area. King (1987) also found the same relationship
through finite element simulation. Take spherical indentation as an example,
differentiating Eq. (25) with respect to h leads to
h
v
R
dh
dp
S
−
= =
1
4μ
, (213)
where S is the contact stiffness. Considering r = Rh 2
0 , we have
(1 )
2
2 v
AE
dh
dp
S
−
= =
. (214)
Pharr, Oliver and Brotzen (1992) demonstrated that above equation is the general
relationship between contact stiffness, contact area and modulus for all kinds of
14
axisymmetric rigid indenters. Based on the above equation, Oliver and Pharr (1992)
proposed an improved method to measure Young’s modulus and hardness. In their
method, the loaddisplacement curve at initial unloading is fitted with a power law
equation. The power law equation is used to calculate the stiffness S. The contact areas
which is function of the contact radius is determined by relating contact radius hc to the
indentation displacement h, based on Sneddon’s solution to axisymmetric indentation
problem. The contact area, taken as function of contact depth after indenter shape
calibration, has such form as follows
L
1/ 8
4
1/ 4
3
1/ 2
1 2
2
0 c c c c c A = a h +a h + a h + a h + a h (215)
The Young’s modulus is then calculated through Eq. (214) after the contact area and
contact stiffness are known. The method documented by Oliver and Pharr is regarded as a
standard technique to measure elastic properties, and has been frequently quoted in
nanoindentation. In this dissertation, some part of this method will also be applied and
compared for indentation on polymers.
2.2 Local stressstrain analysis of indentation problem
Sneddon (1965) found the relation of indentation load and depth, and he also gave the
pressure distribution in indentation direction. In this section, equations for the local
surface stress of sample are derived based on displacement field proposed by Sneddon on
elastic indentation problem.
15
r
hc
Z
O
h
a
Fig. 22 Elastic indentation problem
Boundary condition:
(r,0) = 0 rz ,
u (r,0) h f (r / a) z = − , 0 r a (216)
(r,0) = 0 zz , r> a
where f(r/a) is the shape function, expressed as the depth of the tip to the cross section of
radius of r.
Sneddon showed that displacement components fields could be specified as “dual integral
equations” as follows
[(1 2 ) ( ) ]
2(1 )
( , ) 1
1
z
r v z a e
v
a
u r z − − − −
−
= − h ,
[(2(1 ) ) ( ) ]
2(1 )
( , ) 1
0
z
z v z a e
v
a
u r z − + − −
−
= h , (217)
( = 0 u )
where [ f ( , z)] f ( , z)J ( r)d
0
h = , is Hankel transform of order of ( =0,1 in
this case).
And among these equations, ( ) can be represented as
16
= 1
0
( ) (t) cos( t)dt . (218)
Sneddon (1965) obtained the function (t) for different kinds of indenter. For conical
indenter,
(1 )
2
( ) t
h
t = −
. (219)
Substituting (219) into (218), one has
(1 cos )
2 1
( ) 2
= − h
. (220)
Substituting (220) into (3016), one has
[(1 2 ) ( ) ]
2(1 )
( ,0) 1
1
z
r v z a e
v
a
u r − − − −
−
= − h
−
−
= − 1
0 0
1 ( ) ( ) cos
2(1 )
(1 2 )
J r d t atdt
v
a v
−
= − 1
0 0
1 ( ) ( ) cos
2(1 )
J x d t tdt
v
a ( ,
a
r
x = and x t)
t dt
h
v x
v
(1 )
1 2
2(1 )
1 2 1
0
−
−
−
= −
r
ah
v
v
2(1 )
2 1
−
−
= , (221)
u (r,0) h f (r / a) h r tan z = − = − , (222)
= 0 u . (223)
Next, strain is determined from Eqs. (217) and (221). Because of axisymmetric
characteristic for nanoindentation problems, the strain field can be computed as
following:
17
z
uz
z
= ;
r
ur
r
= ;
r
u= r . (224)
Then
( ) v z e d
z
J r a
v
a v
z
u z z
z ( ) ( ) {[2 1 ] }
2(1 )
(1 2 )
0
0
−
− +
−
−
= −
= , (225)
J r a d
v
a v
z ( ) ( )
2(1 )
(2 1)
0
0 0
= −
−
=
[ ( )]
2(1 )
2 1
0 h
v a
v
−
−
= . (226)
From Sneddon’s solution for (r,0) z ,
[ ( )]
(1 )
( ,0) 0
μ
h
a v
r z −
= −
cosh ( / )
(1 )
2 1 a r
a v
h −
−
= −
μ
, (227)
we can derive
cosh ( / )
(1 )
(2 1) 1 a r
v a
v h
z
−
−
−
=
, (z=0) (228)
2 2(1 )
1 2
r
ah
v
v
r
ur
r −
−
=
= , (z=0) (229)
2 2(1 )
1 2
r
ah
v
v
r
ur
−
−
= = − . (z=0) (230)
Substituting Eqs. (228), (229) and (230) into Stress Strain relationship, one can obtain
stresses at the surface (z = 0),
)
1 2
(
1 r
u
v
v
v
E r
r kk
+
+ −
=
]
2
1
cosh ( )
1 2
[
1
(2 1)
2
1
2 r
ah
r
a
a
h
v
v
v
E v −
− −
−
= −
, (231)
18
)
1 2
(
1 r
u
v
v
v
E r
kk +
+ −
=
]
2
1
cosh ( )
1 2
[
1
(2 1)
2
1
2 r
ah
r
a
a
h
v
v
v
E v +
− −
−
= −
, (232)
cosh ( / )
(1 )
2 1 a r
a v
h
z
−
−
= −
μ
; = rz = r = 0 z . (233)
According to Sneddon, for conical indenter, there are
μ
tan
1
2
v
a
P
−
= , tan
2
1
h = a .
Using these relationships, we can rewrite Eqs. (231) and (232) in another form,
P
r
v
r
a
v
v r 2
1
2
(1 2 )
tan cosh ( )
1
μ
−
" +
−
= − − , (234)
P
r
v
r
a
v
v 2
1
2
(1 2 )
tan cosh ( )
1
μ
−
" −
−
= − − . (235)
These and Sneddon’s result for z ,
cosh ( / )
(1 )
2 1 a r
a v
h
z
−
−
= −
μ
, (236)
give all the stress components in elastic contact problem at the interface of the contact
surface and free surface, r = a . These stress components are
P
a
v
rr r 2 2
(1 2 )
'
−
= =
, (237)
P
a
v
2 2
(1 2 )
'
−
= = −
, (238)
'= = 0 z z . (239)
2.3 Linear viscoelasticity
The stressstrain relation can be described by convolution integral based on Boltzman
19
superposition principle, which takes stress (strain) prehistory into consideration. The
linearly viscoelastic constitutive law is
d
d
d
t D t
t
kl
ij ijkl
−
= −
( )
( ) ( ) . (240)
where ijkl D are components of creep tensor, ij and kl are shear strains and shear
stresses, respectively. For isotropic materials, the deviatoric stress strain relations are
d
d
dS
e t J t
t
ij
ij
−
= −
( )
( )
2
1
( ) , (241)
and
μ d
d
de
S t t
t
ij
ij
−
= −
( )
( ) 2 ( ) , (242)
where J and μ are creep compliance and relaxation modulus, respectively.
If time is defined as positive, one has
d
d
dS
e t J t S J t
t
ij
ij ij ij = + + −
0
( )
( )
2
1
( ) (0 )
2
1
( ) (243)
and
μ μ d
d
de
S t t e t
t
ij
ij ij
−
= + + −
( )
( ) 2 ( ) (0 ) 2 ( ) . (244)
2.4 Linearly viscoelastic indentation problem
Indentation into half space of a workpiece is a three dimensional problem. Because
the contact stresses and strains are highly localized close to the contact area, and their
magnitude decreased rapidly with distance from the point of contact, the deformation
field is inhomogeneous. Therefore, with complexity of stress and strain distribution under
the indenter, it is not practical to solve a viscoelastic indentation problem directly based
20
on the constitutive equations, such as Eqs. (240) and (241). It is noted that load and
displacement are the only data output from nanoindentation. Thus, it is convenient to
characterize the linearly viscoelastic behavior by examining loaddisplacement
relationship.
For viscoelastic contact problems, Lee (1955) found his solution by applying inverse
Laplace transformation of the corresponding elastic problem. However, such a method is
valid only when the boundary conditions can be prescribed. For nanoindentation of
viscoelastic materials, both the traction boundary and displacement boundary are
changing with time as the contact between the indenter and the workpiece changes with
time. It is not possible to predict explicitly the history of both traction boundary and
displacement boundary. Therefore direct application of the correspondence principle is
not appropriate for solving viscoelastic indention problem. Lee and Radok (1960) later
proposed an approach applicable for viscoelastic contact problems with moving boundary
conditions. They showed that the Hertzian contact problem in viscoelasticity could be
solved by replacing the elastic constants in elastic problem with integral operators under
condition of nondecreasing contact area, such that the corresponding viscoelastic
problem could be expressed in hereditary form. Using the Hertzian solution to elastic
indentation problem and assuming that the viscoelastic material is incompressible (Eq.2
5), the loaddisplacement relation follows for a spherical indenter,
( )
3
8
( )
2
1 3 / 2
0
d Rh t
d
dP
J t
t − # =
#
# . (245)
In this dissertation, Lee and Radok’s hereditary operator based on elastic indentation
solution will be applied to develop a consensus method to measure viscoelastic properties
of polymers.
21
CHAPTER III.
MEASUREMENTS OF CREEP COMPLIANCE (TIMEDOMAIN)
USING NANOINDENTATION
3.1 Analytical prerequisites
In this section a method to measure local surface creep compliance for linearly
viscoelastic materials is proposed and validated.
As introduced before, the loaddisplacement relationship for spherical indentation
problem, i.e., Hertz problem is:
2
3
3(1 )
8
h
R
P
$
μ
−
= , (31)
where μ is shear modulus, v Poisson’s ratio, R the radius of the spherical indenter, and r0
contact radius. Applying Lee and Radok’s method, one has
$
d
d
dP
J t
R
h t
t
−
%&
'
()
*
−
−
=
( )
( )
8
3(1 )
( ) 3 / 2 , (32)
Under a constant rate loading history, ( ) ( ) 0 P t = v tH t , with 0 v being loading rate, Eq. (3
2) becomes
$
J d
R
v
h t
t
−
=
0
3 / 2 0 ( )
8
3(1 )
( ) , (33)
22
where 0 v is loading rate. Differentiation of Eq.(33) with respect to time yields
dt
dh
v
R h t
J t
0 (1 )
4 ( )
( )
−$
= . (34)
Sneddon’s solution for elastic conical indentation problem, in terms of loaddisplacement
relationship is
2
(1 )
4 cot
h
v
P
−
=
μ
(35)
where S is the angle between the cone generator and the substrate plane. Again applying
Lee and Radok’s method to Eq.(35) leads to:
$
d
d
dP
h t J t
t
−
%&
'
()
*
−
−
=
( )
( )
4cot
(1 )
( ) 2 (36)
Under a constant rate loading, ( ) ( ) 0 P t = v tH t , with H(t) being Heaviside function,
finally Eq.(36) becomes
dt
dh t
v
h t
J t
( )
(1 ) tan
8 ( )
( )
0 −$
= (37)
Eq.(33) and Eq.(37) will be used for the computation of creep compliance in terms of
derivative of displacement with respect to time under constant rate loading. Due to the
fact that data from for displacement nanoindentation experiments are usually scattered,
the derivative of displacement with respect to time based on experimental data can induce
some error, even if the related curve is fitted. An alternative approach is proposed next.
The general representation of the creep function based on the generalized Kelvin model is
( ) (1 )
1
0 +=
−
= + −
N
i
%
t
i
J t J J e i , (38)
23
where J , J , , J N 0 1 " " " are compliance numbers, and N % , % , , % 1 2 " " " are retardation times.
For the Berkovich indenter, substituting Eq.(38) into Eq. (36) one has
+ +
=
−
=
+ − −
−
=
N
i
%
t
i i
N
i
i
v J J t J % e i
ctg &
) (
h t
1 1
0 0
2 [( ) (1 )]
4
(1 )
( ) . (39)
Considering P t v0t ( ) = , Eq.(39) can be rewritten as
+ +
=
−
=
+ − −
−
=
N
i
v %
P t
i i
N
i
i
J J P t J v % e i
ctg &
) (
h t
1
( )
0
1
0
2 [( ) ( ) ( )(1 )]
4
(1 )
( ) 0 . (310)
If we fit Eq. (310) into the experimentally measured loaddisplacement curve using the
least square correlation, we can find a set of bestfit parameters N J , J , , J 0 1 " " " and
N % , % , , % 1 2 " " " . We may then substitute these constants to Eq. (38) to determine the creep
function when the Berkovich indenter is used in nanoindentation.
The same method for data reduction to determine J(t) can be applied to a spherical
indenter. For a spherical indenter, substitution of Eq. (35) into Eq. (33) leads to
+ +
=
−
=
+ − −
−
=
N
i
%
t
i i
N
i
i
J J t J % e i
R
( v
h t
1 1
0
3 / 2 0 [( ) (1 )]
8
3(1 )
( ) , (311)
Since P t v0t ( ) = , Eq.(311) can be rewritten as
+ +
=
−
=
+ − −
−
=
N
i
v %
P t
i i
N
i
i
J J P t J v % e i
R
(
h t
1
( )
0
1
0
3 / 2 [( ) ( ) ( )(1 )]
8
3(1 )
( ) 0 . (312)
Similar to the approach for a conical indenter, we can fit Eq. (312) into the
experimentally measured loaddisplacement curve to find a set of the bestfit parameters
N J , J , , J 0 1 " " " and N % , % , , % 1 2 " " " , we then substitute these parameters into Eq. (38) to
determine the creep function when a spherical indenter is used in nanoindentation.
24
It is noted that applicability of the hereditary integral operator provided by Lee and
Radok (1960) as shown in Eqs. (33) and (36) is based on the condition that contact area
between indenter and workpiece is nondecreasing with time. It should be pointed out
that in Eqs. (33) and (36) the Poisson’s ratio is assumed to be constant.
3.2 A corrected method to obtain creep function under a step loading
Section 3.1 details the determination of creep function from indentation under ramp
loading. Alternatively, under step loading the creep function could be also obtained. For
a step loading suddenly applied in indentation, it could be expressed in terms of
Heaviside function:
( ) ( ) 0 P t = P H t . (313)
Substituting it into Eq.(36) , one can derive the creep function for Berkovich indenter:
.
(1 ) tan
4 ( )
( )
0
2
) ( P &
h t
J t
−
= (314)
Also substituting it into Eq.(32) , one has for spherical indentation:
.
3(1 )
8 ( )
( )
0
3 / 2
( P
Rh t
J t
−
= (315)
Eqs. (314) and (315) are derived for theoretical step loading, however, such ideal step
loading could never be realized in practice. On one hand, the sudden increase of loading
will take some certain time, even though it is very short. On the other hand, even
seemingly infinitesimal time of increase of sudden load will cause unfavorable impact on
the instrument, which could induce error for experimental data. Thus, instead the creep
test is implemented by applying ramp loading with short rise time (usually 1s in this
25
study) followed by constant loading hold for comparatively long time. According to
conventional creep test, the beginning data for 10 times of the rising time are not reliable
for computation, which means in creep indentation experiment, the initial data of 11 s
cannot be taken for determination of creep function of linearly viscoelastic materials, i.e.,
the creep function for 11 s in the beginning is lost. Nonetheless, such a period of time is
a considerable portion, if the whole time scale for experiment is not large. Lee and
Knauss (2000) proposed a method to compute shear modulus for that period time of
interest. They used Boltzman superposition principle to reformulate the relationship
between stress and strain. To avoid the loss of the accurate data, this study took a method
similar to that used by Lee and Knauss.
Fig. 31 A ramp loading history
The work presented by Lee and Knauss is on the data correction for uniaxial stress
state, while indentation problem as discussed herein is a three dimensional stress state.
Formulas need to be derived for data correction to determine the creep function using
loaddisplacement relation in a viscoelastic indentation problem. As shown in Fig.(31),
based on Boltzman superposition principle, a realistic loading, can be considered as the
superposition of two different loading histories
t0
P0
t0 t t t
P P P
26
( ) ( ) ( ) ( ) ( ) ( ) 1 2 0 0 0 0 P t = P t − P t = v tH t − v t − t H t − t , (316)
where 0 v is the ramp loading rate.
For a conical indenter, substituting ( ) ( ) ( ) ( ) 0 0 0 0 P t = v tH t − v t − t H t − t into Eq.(36) ,
one has: When t< 0 t ,
$
J d
v
h t
t
−
=
0
2 0 ( )
4
(1 ) tan
( ) ; (317)
and when t> 0 t ,
[ ( ) ( ) ]
4
(1 ) tan
( )
0 0
2 0
$
J t d J t d
v
h t
t
t
t
− − −
−
=
[ ( ) ( ) ]
4
(1 ) tan 0
0 0
0
$
J d J d
v t t t
−
−
−
= . (318)
Differentiation of Eqs.(317) and (318) with respect to time t yields
dt
dh t
v
h t
J t
( )
(1 ) tan
8 ( )
( )
0 −$
= ( ) 0 t < t (319)
dt
dh t
v
h t
J t t J t
( )
(1 ) tan
8 ( )
( ) ( )
0
0 −$
− = − ( ) 0 t t (320)
Similarly, for a spherical indenter, the following results are obtained:
dt
dh t
v
Rh t
J t
( )
(1 )
4 ( )
( )
0
1/ 2
−$
= ( ) 0 t < t (321)
dt
dh t
v
Rh t
J t t J t
( )
(1 )
4 ( )
( ) ( )
0
1/ 2
0 −$
− = − ( ) 0 t t (322)
Therefore, the procedure of data correction could be considered as reversed computation
started at some point, for example, ten times of rise time. For a conical indenter, using
27
Eqs. (319) and (320), the data of the creep function determined by Eq. (314) can be
corrected through the following steps:
(i) For 0 0 kt t (k +1)t , compute ( ) 0 J t − t at 0 0 t = kt + m*t by Eq. (320) and
result of J(t) calculated with Eq. (314), where, is some sufficiently small
number, 0 < m 1/, , and k is a positive integer, usually k 10 .
(ii) For 0 0 (k −1)t t kt , compute ( ) 0 J t − t at 0 0 t = (k −1)t + m*t by Eq. (320)
and result from (i).
(iii) Repeat the same step as (ii) for 0 0 (n −1)t t nt , where n = k −1, k − 2,L,3 .
(iv) For 0 , 0 t < t compute J(t) by Eq. (319).
For a spherical indenter, the same method can be used to correct the initial part of creep
function. Simply replace Eqs. (314), (319) and (320) in (i)(iv) for a conical indenter
by Eqs. (315), (321) and (322), respectively.
3.3 Experiments
An MTS Nano Indenter XP system was employed in nanoindentation, where both
Berkovich and spherical indenters were used. All the experiments were started after the
indenter drift rate due to environment noise reached within 0.05nm/s. The room
temperature and humidity were carefully monitored and maintained to be 22.5 oC and
50% respectively.
Two bulk polymers, Polycarbonate (PC) and Polymethyl Methacrylate (PMMA) were
chosen as test materials. The glass transition temperature for PC and PMMA are 145 oC
and 105 oC respectively. They were annealed for two hours at the temperature 5 oC above
28
their glass transition temperatures, and cooled down to room temperature at rate of 5
oC/hr. After annealing, the sample materials were stored at bell jar for 72 hours before the
experiments were carried out.
3.4 Result and discussion
As shown in Fig.32, a ramp loading history and a step loading history were applied
in the experiments. The resulted loaddisplacement curves at ramp loading for both
Berkovich indenter and spherical indenter are shown in Fig.33.
(a) Constant rate loading (b) Realistic step loading
Fig. 32 Constant loading rate history and step loading history
Because the measurement of creep function is based on linear viscoelasticity, it’s
critical that overall deformation in the course of indentation is within the limit of
linearity. It is assumed that when the indentation depth is small enough, the displacement
will be fully recovered some time after complete unloading, and it could be inferred that
Time (s)
mN)
0 100 200 300 400
0
1
2
3
4
5
6
7
8
Load (mN)
100 200 300
2.5
5
7.5
Load (mN)
100 200 300
2.5
5
7.5
Time (s)
0 100 200 300 400
0
5
7.5
29
the test materials should deform within the limit of viscoelastic linearity. One
straightforward way to determine the linearity is to observe the impression after removal
of the loading. After experiments, the material sample surface was observed with help of
SEM (Scanning Electron Microscopy). It was found that for PMMA, when the depth was
below 780 nm, no impression was observed. And also it was observed that for PC no
impression left when the depth below 1123 nm. Thus approximately, the depth limit of
linearity for PMMA and PC could be considered as 780 nm and 1123 nm respectively.
For experiment at ramp loading, the method in terms of derivative of displacement
with respective to time and exponential fitting of loaddisplacement curve, i.e. Eqs.(34)
and (37), were applied to retrieve the creep function for PMMA and PC. For curve
fitting method, the loaddisplacement curve was fitted at displacement less than limit of
linearity. Fig 34 shows the fitted curves for PMMA and PC.
0 50 100 150 200 250 300
0
0.1
0.2
0.3
0.4
0.5
0.6
PMMA
Polycarbonate
0 500 1000 1500
0
1
2
3
4
5
6
PMMA
Polycarbonate
(a) Berkovich indenter (b) Spherical indenter
Fig. 33 Loaddisplacement curves for Berkovich indenter and spherical indenter
For experiment at step loading, the method introduced in Sec. 3.1.3 is used to correct
data of creep function. Because of “fading effect” of viscoelastic materials, the creep
Displacement (nm)
Load (mN)
Load (mN)
Displacement (nm)
30
function after enough long time (here 20s) was computed using Eqs. (314) and (315),
while the initial 20s of creep compliance data were computed from a reversed procedure
described in Sec.3.2.
The final results for creep function from ramp loading experiment and step loading
experiments are shown in Fig.35, where they were compared with conventional data.
Displacement (nm)
Load (mN)
0 400 800 1200 1600 2000
0
4
8
12
16
20 Experiment
Fit
Displacement (nm)
Load (mN)
0 500 1000 1500 2000 2500
0
4
8
12
16
20 Experiment
Fit
(a) PC using Berkovich indenter (b) PMMA using Berkovich indenter
Displacement (nm)
Load (mN)
0 50 100 150 200
0
0.1
0.2
0.3
0.4
0.5
0.6
Experiment
Fit
Displacement (nm)
Load (mN)
0 50 100 150 200 250 300
0
0.1
0.2
0.3
0.4
0.5
0.6
Experiment
Fit
(c) PC using spherical indenter (d) PMMA using spherical indenter
Fig. 34 Experiment curve and fitting curve for loaddisplacement
31
Fig.35 shows that the measured creep compliance of represented solid polymers agree
with well with the conventional data. Thus, within linear viscoelasticity, the introduced
methods to compute creep function are validated to be appropriate for viscoelastic
materials.
Time (s)
Creep Compliance (1/GPa)
0 50 100 150 200 250 300
0
0.2
0.4
0.6
0.8
1
1.2
Conventional J
Ramp Loaddh/dp
Ramp LoadFit
Step Load
Time (s)
Creep Compliance (1/GPa)
0 10 20 30 40 50 60 70
0
0.3
0.6
0.9
1.2
1.5
1.8
Conventional J
Ramp Loaddh/dp
Ramp LoadFit
Step Load
(a) PMMA using Berkovich indenter (b) PC using Berkovich indenter
Time (s)
Creep Compliance (1/GPa)
0 10 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1
1.2
Conventional J
Ramp Loaddh/dp
Ramp LoadFit
Step Load
Time (s)
Creep Compliance (1/GPa)
0 10 20 30 40 50 60 70
0
0.3
0.6
0.9
1.2
1.5
1.8
Conventional J
Ramp Loaddh/dp
Ramp LoadFit
Step Load
(c)PMMA using spherical indenter (d) PC using spherical indenter
Fig. 35 Creep compliance measured from nanoindentation
32
CHAPTER IV
MEASUREMENTS OF COMPLEX CREEP COMPLIANCE (FREQUENCY
DOMAIN) USING NANOINDENTATION
4.1 Introduction
Methods for measuring the Young's modulus have been very well established for
timeindependent materials. Based on the assumption that unloading in the loaddisplacement
curve induces only elastic recovery, Oliver and Pharr (1992) pioneered a
method to determine basic material properties such as the Young’s modulus. The method
is based on Sneddon's solution (1965) for the relationship between the load and
displacement for an axisymmetric indenter indenting into a halfspace composed of a
linearly elastic, isotropic and homogeneous material. While the methods work well for
timeindependent materials (metals, etc.), applying the methods directly to viscoelastic
materials has experienced problems. For example, the unloading curve in viscoelastic
materials sometimes has a negative slope (OyenTiesma et al., 2001), under situations in
which a small unloading rate and a relatively high load were used for a material with
pronounced viscoelastic effects. Some work has been done in recent years to improve the
methods proposed by Oliver and Pharr (1992) to determine the Young's modulus, or the
Young's relaxation modulus. Cheng et al. (2000) derived the analytical solutions for
33
linear viscoelastic deformation, and provided a method to measure viscoelastic properties
using a flatpunch indenter. Lu, et al. (2003) proposed methods to measure the creep
compliance of solid polymers using either Berkovich or spherical indenter. Hutcheson
and Mckenna (2005) analyzed the interaction between nanosphere particles and polymer
surfaces using viscoelastic contact analysis by considering a timedependent Poisson’s
function.
For a viscoelastic material, in addition to the representation of viscoelastic properties
in the timedomain, the material properties can also be represented by complex material
functions in the frequencydomain. A method was proposed by Loubet et al. (1995) for
the computation of the complex modulus of viscoelastic materials. The method uses data
acquired from an MTS Nano Indenter XP System installed with a Continuous Stiffness
Module (CSM). The CSM allows cyclic excitation in load or displacement and the
recording of the resulting displacement or load (Lucas, Oliver, and Swindeman, 1998).
The indentation displacement response and the outofphase angle between the applied
harmonic force and the corresponding harmonic displacement are measured continuously
at a given excitation frequency. Loubet et al. presented the following equations to
compute the complex modulus ( ) E*
:
( ) ' " * E
= E +iE , with
A
S
E
2
'
= and
A
C
E
2
"
= , (41)
where E' and E" are the uniaxial storage modulus and the loss modulus, respectively,
S the contact stiffness, C the damping coefficient, and A the contact area between the
indenter and the workpiece. This method was used to measure the complex modulus of
polyisoprene. In an effort to examine the formulas in Eq. (41), we conducted
34
experiments using an MTS Nano Indenter XP system with CSM, and used the formulas
in Eq. (41) to compute the complex modulus of polycarbonate (PC) and polymethyl
methacrylate (PMMA) at 75Hz. Data are computed at all times at this frequency, and
plotted in Fig. 41, but only steady state values represent the complex viscoelastic
function. Also shown in Fig. 41 are the conventional data measured from Dynamic
Mechanical Analysis (DMA) (for details, please see Sections 4.3 & 4.4) for the same
batch of PC and PMMA. The uniaxial storage modulus of PC measured by DMA at 75
Hz is 2.29 GPa. However, the storage modulus computed using Eq. (41) is much higher
than this value, indicating the difficulty associated with the method described in Eq. (41)
for measurement of storage modulus for PC. Similar problem is evident for PMMA. Also
shown in these figures are storage modulus data measured by the proposed method that
will be discussed in Section 4.4.
Time (s)
Storage Modulus (GPa)
50 100 150
0
1
2
3
4
Nanoindentation (New)
Conventional (DMA)
Nanoindentation (Previous)
Time (s)
Storage Modulus (GPa)
50 100 150
0
1
2
3
4
5
6
7
8
Nanoindentation (New)
Conventional (DMA)
Nanoindentation (Previous)
(a) (b)
Fig. 41 Comparison of storage compliance at 75 Hz computed by two methods (a) PC.
(b) PMMA.
35
This study is intended to develop a method to measure the complex viscoelastic
functions of timedependent materials in the frequencydomain using nanoindentation
with a spherical indenter. Based on solutions for the indentation of an axisymmetric
indenter into a linearly elastic material, viscoelastic indentation under a timeharmonic
loading condition is analyzed using a hereditary integral operator as proposed by Lee and
Radok (1960). Formulas are derived to process the amplitudes of load and displacement
as well as the outofphase angle between load and displacement to determine the storage
and loss parts of the complex compliance (or modulus) function using a spherical
indenter. The LeeRadok approach is applicable to situations where the contact area
between the nanoindenter and the workpiece does not decrease. When the condition of
nondecreasing contact area is not satisfied, Ting’s approach (1966) is used to estimate
the difference between the approximation from LeeRadok approach and the solution
obtained from the Ting approach. Dynamic nanoindentation tests were conducted on PC
and PMMA to determine the complex compliance, and results are compared with data
obtained from DMA on the identical materials to validate the method presented.
4.2 Theoretical background
In this section, we present derivation of the formulas for the computation of complex
compliance for a linearly viscoelastic material. The formula for the complex modulus is
also presented, as it is simply the reciprocal of the complex compliance. Formulations
will be given for a spherical indenter that will be used in experimental verification in this
study.
36
R
h
Z= 0 a
Z
(a) (b)
Fig. 4 2 Geometry of the spherical indenter. (a) Schematic diagram of the indenter. (b) A
TEM image of the spherical indenter tip.
Fig. 42 shows the geometry of a spherical indenter. We first consider the problem of a
spherical indenter indenting into a half space composed of a homogenous, linearly elastic,
isotropic material. The diamond indenter is assumed to be a rigid because of the huge
difference in the Young’s modulus between the indenter and polymer samples; and the
material occupies the half space (z X 0). The spherical indenter has a tip radius R. Based
on the Hertzian solution, under the condition that the ratio of indentation depth to radius
of indenter is not higher than 0.16 (Giannakopoulos, 2000), the relation between the
applied indentation load and the indentation displacement can be expressed by (Hertz,
1881, Ling, 2002)
3 / 2
3(1 )
8
Gh
R
P
−$
= , (42)
R 3.4 Ym
37
where G is the shear modulus, $ the Poisson’s ratio, P the applied indentation load, and
h the indentation displacement.
For the nanoindentation of a spherical indenter into a viscoelastic material, we
assume a constant Poisson’s ratio. During nanoindentation experiments in which a
relatively short time (such as ~250 s used in this study) is involved, the Poisson's ratio
(Lu et al., 1997) does not change significantly for some polymers in the glassy state, such
as PMMA. Thus it is assumed that a constant Poisson's ratio will not cause much error in
the complex compliance data.
For a half space composed of a linearly viscoelastic material, we consider first the
case in which the contact area between the indenter tip and the work material is nondecreasing.
The condition for nondecreasing contact area will be established at the end
of the section. Using the hereditary integral operator proposed by Lee and Radok (1960)
in Eq. (42), the indentation loaddisplacement relation is represented by
3/ 2 3(1 ) ( )
( ) ( )
8
t dP
h t J t d
R d
$
−
−
= − , (43)
where J (t) is the creep compliance function in shear in the timedomain, P(t) = 0 for
t < 0 .
Consider a sinusoidal nanoindentation load superimposed on a step loading,
represented by
P t P H t P t m ( ) ( ) sin
0 = +  , (44)
where H(t) is the Heaviside unit step function, Pm is the carrier load, or main load, [P0 is
the amplitude of the harmonic load. Eq. (44) implies P(t) = 0 for t < 0 .
Inserting Eq. (44) into Eq. (43), we have
38
3/ 2
0
0
3(1 )
( ) [ ( ) ( ) cos ].
8
t
m h t P J t P J t d
R
$
−
= +  − (45)
The contact radius is a(t) = Rh(t) . Considering that the complex compliance is defined
after the harmonic response has reached a steady state (or equivalently t . ), we have
{ ( ) [ '( ) sin "( ) cos ]}
8
3(1 )
( ) 0
3 / 2 P J t P J t J t
R
h t m
$
+  −
−
= (46)
where
0
J '(
)
J (t) sin
tdt
= and
0
J "(
)
J (t) cos
tdt
= − are storage compliance
and loss compliance in shear, respectively. Note that the complex compliance in shear is
( ) '( ) "( ) J *
= J
− iJ
. On the other hand, the total displacement as output from a
nanoindenter is expressed as
( ) ( ) sin( ) 0 h t = h t +h
t −/ m , (47)
where h (t) m is the carrier displacement, and / is the outofphase angle between the
applied harmonic force and displacement. This representation is based on the fact that
displacement is behind of load in phase in dynamic nanoindentation. 0 h is typically of
the order of a few nm while hm(t) is of the order of a few hundreds nm under step loading
so that 0 h << h (t) m , Eq. (47) leads to
( ) sin cos ( ),
2
3
( ) cos sin
2
3
( ) ( ) 0 0
1/ 2
0
3 / 2 3 / 2 1/ 2 h t h t h t h t h t h t o h m m m = +  /
−  /
+  (48)
where ( ) 0 o h represents high order terms of 0 h , and they are negligible under
condition of 0 h << h (t) m . Comparing Eq. (46) with Eq. (48), we find that
( )
8
3(1 )
( ) 3/ 2 P J t
R
h t m m
−$
= , (49)
39
/


$
cos
( )
1
4
' ( )
0
0
1/ 2
P
R h t h
J m
−
= , and /


$
sin
( )
1
4
" ( )
0
0
1/ 2
P
R h t h
J m
−
= . (410)
Under the loading condition in which a small sinusoidal load is superimposed upon a
constant rate loading, i.e.,
P(t) v t P sin
t 0 0 = +  , (411)
where 0 v is the loading rate, following similar procedures as in deriving Eq. (410), the
formulas to determine complex compliance can also be derived under the condition that
the time t has evolved to a value such that h (t) m >> 0 h .
Substituting Eq. (411) into Eq. (43) for the spherical indenter, we have
3/ 2 [ ]
0 0
0
3(1 )
( ) ( ) '( ) sin "( ) cos
8
t
h t v J t d P J t J t
R
$
− 02 12
= 3 − +  − 4
25 26
. (412)
Comparing Eq. (412) with Eq. (48), the same formulas as in Eq. (410) for the complex
compliance can be derived for a small oscillatory load superimposed upon a constant rate
loading.
Eq. (410) is used to determine the complex compliance in shear from
nanoindentation. The uniaxial complex compliance,D(
) , can be computed from
2[1 ( )]
'( ) "( )
( ) ' ( ) "( ) *
*
$
+
−
= − = J iJ
D D iD , (413)
where D' (
) and D"(
) are uniaxial storage modulus and loss modulus, respectively,
( ) $ *
the complex Poisson’s ratio. Assuming that ( ) $ *
is a constant during shorttime
nanoindentation tests for glassy polymers, from Eqs. (410) and (13), D' (
) and
D"(
) can be computed by
40
cos ,
( )
1
2
' ( )
0
0
1/ 2
2 /
$
P
R h t h
D m


−
= and sin .
( )
1
2
"( )
0
0
1/ 2
2 /
$
P
R h t h
D m


−
= (414)
Similarly, the complex modulus in shear can also be determined by ( ) 1/ ( ) G*
= J *
,
and the uniaxial complex modulus can be computed by ( ) 1/ ( ) E*
= D*
.
It should be noted that Eq. (43) is valid only if the indentation contact area is nondecreasing
(Lee and Radok, 1960). Under the oscillatory loading condition, indentation
can possibly induce decreasing contact area, in which case, the LeeRadok integral
operator will cause a residual surface traction at points not in contact at current time but
formerly within the contact region, thus violating the boundary condition that the surface
traction should vanish outside the contact region. In the case of arbitrary contact area
history, Ting (1966) developed an analytical approach to solve the problem of
axisymmetric viscoelastic indentation by a rigid indenter. The Ting approach leads to the
same results as those derived from the LeeRadok approach when the contact area is nondecreasing.
The Ting approach, however, is necessary in the case where decreasing
contact area occurs in nanoindentation.
We next provide a condition under which nondecreasing contact area is maintained
so that the solution derived from the LeeRadok approach is valid. In terms of Eq. (48)
and a(t) = Rh(t) , for a small harmonic loading superimposed on a constant rate
loading, if
0 0 v P , as seen from Eq. (411) for a harmonic loading superimposed on a
constant rate loading, the nondecreasing load leads to the nondecreasing contact area in
the entire indentation history. For a harmonic loading superimposed on a step loading,
when 0 h h m
&  , from Eq. (47) the contact area will be nondecreasing during the
whole process; as the frequency exceeds the critical value 0 h h c m
= &  , the contact area
41
increases and decreases with time as a result of the applied harmonic load so that the Ting
approach should be adopted. Nevertheless, as will be discussed in Section 4.2, under
certain condition when c
> , the solutions derived from the methods by Lee & Radok
and Ting are very close, even though the solution from the LeeRadok approach is not
justified. Since a closedform solution derived from the LeeRadok approach exists,
while only numerical solution can be obtained using the Ting approach, the formulas
derived for a harmonic superimposed on a step loading from the LeeRadok approach
could be used to estimate the complex viscoelastic functions in the regime of linear
viscoelasticity.
4.3 Experiments
We conducted two independent tests, namely nanoindentation and DMA tests, to
find the complex viscoelastic functions of the same materials. The results would be
compared to examine the measurement technique by nanoindentation. We describe in
this section first DMA experiments and then nanoindentation experiments.
4.3.1 DMA experiments
The conventional data of complex compliance were obtained by DMA tests. Dynamic
Mechanical Analyzer, model RSA (Rheometric Scientific), was employed in the
measurements of complex compliance. In DMA threepoint bending tests were conducted
on both PC and PMMA, which have the dimensions of 50 × 13 × 1.7 mm and 50 × 13 ×
1.4 mm, respectively. The PC material was made by GE Plastics. The PMMA was made
by Rhom and Haas, which is the same batch of materials as used by Lu et al. (1997). The
42
glass transition temperatures for PMMA and PC are 105 oC and 144 oC, respectively.
Before testing, all samples were annealed for two hours. The annealing temperatures for
PMMA and PC samples were 110 oC and 150 oC, respectively. They were then cooled
down slowly, at a cooling rate of about 5 oC/hr, to room temperature. The annealed
samples were further stored in a container with a constant relative humidity 50%, to
allow the samples to be aged for about 72 hrs prior to DMA tests. These procedures are
necessary to ensure property consistency as the behavior of polymers depends on a
variety of conditions including previous stress history, aging time and moisture
concentration.
For the comparison with the complex compliance data from nanoindentation conducted at
22 oC, the frequency range is 0  260 Hz. However, The DMA can only reach a frequency
up to 16 Hz. In order to extend the frequency range for conventional data of complex
compliance, temperaturefrequency tradeoff was applied (Ferry 1950). To implement
this, DMA tests were performed at selected lower temperatures. The complex compliance
data at these temperatures were shifted to obtain master curves that cover the frequency
range of 0  260 Hz (see Section 4.1 for details).
4.3.2 Nanoindentation experiments
Nanoindentation tests were conducted using an MTS Nano Indenter XP system. The
capacities of indentation depth and load with this system are 500 μm and 500 mN,
respectively. The resolutions of displacement and load are 0.01 nm and 50 nN,
respectively. A spherical indenter with a tip radius 3.4 μm was used on the XP module.
43
Fig. 42(b) shows a TEM micrograph of the spherical indenter tip. The tip is
axisymmetric and has a spherical surface within the depth of indentation (up to ~500 nm)
considered in this study. The same PC and PMMA in DMA tests (see Section 4.3.1) were
used in nanoindentation experiments. The material preparation procedures were identical
to those used in DMA experiments. All nanoindentation tests were performed at 22 oC.
The surface roughness values as measured by an AFM (Digital Instruments Dimension
3100) were 2.859 nm for PC and 2.286 nm for PMMA, respectively. These small surface
roughness values justify the consideration of smooth and flat sample surfaces in
nanoindentation involving indentation depths up to a few hundreds nm; nanoindentation
results under the same conditions were found to be repeatable. In order to reduce the drift
caused by noise and temperature gradient during nanoindentation tests, the indenter
system was enclosed in a chamber. All nanoindentation tests did not start until a thermal
equilibrium state was reached and the drifting of the indenter tip dropped below a set
value, typically 0.05 nm/s.
The CSM implemented in XP module was used to apply the dynamic excitation with
a frequency range of 3  260 Hz. A prescribed harmonic displacement was set before each
experiment. In experiments, the nanoindenter modulates the amplitude of harmonic force
to produce the set target in harmonic displacement, typically with amplitude between a
fraction of a nm and a few nm. After the indenter tip had made contact with the surface of
test sample, the indentation load, depth, harmonic load amplitude, harmonic displacement
amplitude, and outofphase angle between the harmonic load and the displacement were
recorded simultaneously at a sampling rate of five data points per second. The loading
history employed in the dynamic indentation test was small harmonic loading
44
superimposed on a quasistatic loading, i.e., either a step loading or a constant rate
loading. The step loading was implemented by a constant rate load (at a shortrise time,
e.g., 2 s) followed by a constant load. After the harmonic response had reached a steady
state, data were used to determine the complex compliance of the material.
Log (Frequency) (Hz)
Complex Compliance in Shear (1/GPa)
2 1.5 1 0.5 0 0.5 1 1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
PC
PMMA
Storage Compliance in Shear
Loss Compliance in Shear
10 C o
22 C o
10 C o
0 C o
22 C o
10 C o
0 C o
Log(Frequency ) (Hz)
Complex Compliance in Shear (1/GPa)
1 0 1 2 3 4 5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
PC
PMMA
Storage Compliance in Shear
Loss Compliance in Shear
4.4 Results and discussions
4.4.1 DMA results
DMA tests were performed at temperatures 22 oC, 10 oC, 0 oC and 10 oC for PMMA,
and 22 oC, 10 oC and 0 oC, for PC. Fig. 43 shows the storage compliance and loss
compliance in shear for PMMA and PC at different temperatures. As shown in Fig. 43,
the complex compliance of PC does not change much in this range of temperature and
frequency, while the complex compliance of PMMA presents considerable change. Based
Fig. 43 Complex compliance from DMA
tests at different temperatures.
Fig. 44 Master curves of complex
compliance in shear
45
on the curves of complex compliance at different temperatures, very smooth master
curves are obtained using frequencytemperature superposition. The shift factors, in
logarithmic scale referred to 22 oC, Log T a , are 1.6 s and 3.2 s at 10 oC and 0 oC for PC,
respectively; the Log T a for PMMA are 1.2 s, 2.0 s and 2.8 s at 10 oC, 0 oC and 10 oC,
respectively. Fig. 44 shows the master curves of complex compliance of PMMA and PC,
reaching up to 4 10 Hz. The range is wide enough for the comparison with
nanoindentation data. It is observed from Fig. 44 that there is no
 transition within the
frequency range for both PC (Sane and Knauss, 2001, Knauss and Zhu, 2002) and
PMMA (Lu et al., 1997).
4.4.2 Nanoindentation results
Results on the complex compliance from nanoindentation measurements are
presented and discussed in this section. Two types of loading histories were applied in the
indentation tests; they were: (1) a small harmonic load superimposed on a constant rate
loading; and (2) a small harmonic load superimposed on a step loading. We first present
the input and output of nanoindentation tests under the two loading histories.
46
Carrier Displacement (nm)
Carrier Load (mN)
0 200 400 600 800
0
0.5
1
1.5
2
2.5
PMMA
PC
Time (s)
Amplitude of Harmonic Load (uN)
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
PC
PMMA
I
(a) (b)
Time (s)
Amplitude of Harmonic Displacement (nm)
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
PC
PMMA
Time (s)
Phase Angle (Degree)
0 10 20 30 40 50
0
5
10
15
20
25
30
PC
PMMA
(c) (d)
Fig. 45 Nanoindentation output from oscillation on a constant rate loading at 10 Hz. (a)
Carrier loaddepth curve. (b) History of amplitude of harmonic load. (c) Response of
harmonic displacement amplitude. (d) Outofphase angle with correction
47
Time (s)
Contact Radius (um)
160.01 160.015 160.02 160.025
1.132
1.134
1.136
1.138
1.14
LeeRadok Approach
Ting's Approach
I
t1 t
Time (s)
Contact Radius (um)
160.01 160.015 160.02 160.025
1.295
1.296
1.297
1.298
1.299
1.3
1.301
1.302
LeeRadok Approach
Ting's Approach
I
(a) (b)
Fig. 46 Comparison of contact radius results at 75 Hz using the LeeRadok approach and
the Ting approach. (a) Contact radius computed for PMMA under a harmonic load
superimposed on a step loading. (b) Contact radius computed for PC under a harmonic
load superimposed on a step loading.
Constant rate carrier loading was used in the first type of dynamic indentation. When
the loading rate 0 v is greater than
0 P , the contact area will be nondecreasing under
dynamic loading so that the LeeRadok approach is applicable.
The loading rate used for both PC and PMMA is 0.04 mN/s. Fig. 45(a) shows the
quasistatic component of loaddepth curve. Fig. 45(b) shows the harmonic load
amplitude as the input, and Fig. 45(c) shows the harmonic displacement in response to
the harmonic load. The steady values of 0 P for PC and PMMA are 0.445 μN and 0.602
μN, respectively. Therefore the condition,
0 0 v P , is satisfied, resulting in nondecreasing
contact area. It is noted that the condition for nondecreasing contact area can
be satisfied at all frequencies, up to 135 Hz, in nanoindentation with the use of
appropriate loading rate 0 v so that the formulas in Eq. (410) can be applied to find the
48
complex compliance in shear. In addition to the requirements in nondecreasing contact
area, it is necessary to ensure that the harmonic displacement is much smaller than the
carrier displacement. As the carrier displacement increases with time to larger values, for
example, 200 nm for PMMA and PC, the contribution from higher order terms of h0 / m h
is well less than 1% so that the use of Eq. (410) is justified. The outofphase angle
between harmonic load and harmonic displacement is shown in Fig. 45(d).
The second dynamic loading condition is a harmonic load is superimposed on a step
loading. As discussed in Section 4.2, if 0 h h m
&  , the contact area will be nondecreasing,
indicating that when the creep rate is high and the amplitude of harmonic
displacement is small, the nondecreasing contact area condition can be still satisfied at
some frequencies, so that the solutions in Eq. 410 using the LeeRadok hereditary
operator are applicable. For example, with the indentation input shown in Figs. 47(a) and
(b) (details of Fig. 47 will be discussed later), the frequency limits, c
, below which the
condition of nondecreasing area holds, are 2.4 Hz for PC, and 6.5 Hz for PMMA,
respectively. If 0 h h m
> &  , the contact area will decrease. In this case, LeeRadok
approach seems not applicable. However, considering the loading condition in this study,
the resulting harmonic displacement for both PC and PMMA is very small compared to
the carrier displacement, normally, m h 0.006h 0  << . Therefore, the variation in the
contact area between the indenter and the sample surface is always less than ~1% after a
steady state in oscillatory response has reached, so that the effect of change in the contact
area is not significant. To demonstrate this, the viscoelastic indentation problem is solved
numerically using the Ting approach. With a periodical displacement output in the form
of Eq. (47), it suffices to consider only one cycle of the history in the steady state. We
49
present results on contact radius only; other results, such as the indentation displacement,
can be obtained in the similar way. In the first half cycle starting from the valley (lowest
point in a cycle) in the steady state, the contact area is increasing; the Ting approach
gives the same results as those obtained from the LeeRadok approach, that is
3
0
3(1 )
( ) ( ) ( ( ))
8
t R
a t J t d P
$
# #
−
= − (415)
where P(t) is given by Eq. (44). For the second half of the cycle, in which the contact
area is decreasing, the solution derived from the Ting approach is
( ) ( ), 1 a t = a t (416)
where 1 t is obtained by
( ) 1
3
0
8
( ) ( ) ( )
3(1 )
t
R
P t G t # d a #
$
= −
− , (417)
where G(t) is the shear relaxation modulus. Using the creep functions of PC and PMMA,
the contact radius in one cycle can be determined after the sinusoidal response has
reached the steady state. Fig. 46 shows the results of contact radius for PC and PMMA
materials at 75 Hz, under a harmonic load superimposed on a step loading. At 75 Hz, the
frequency limit for nondecreasing contact area condition has been exceeded. The results,
however, indicate that the contact radius computed by the LeeRadok approach correlates
well with those obtained by the Ting approach. The correlation coefficients for PMMA
and PC are 0.939 and 0.993, respectively. At frequencies higher than c
, Eq. (410) and
its variants should be considered as an approximation for the viscoelastic functions in the
frequencydomain.
50
Carrier Displacement (nm)
Carrier Load (mN)
0 100 200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
1.2
1.4
PC
PMMA
Time (s)
Amplitude of Harmonic Load (uN)
0 50 100 150 200 250 300
0
5
10
15
20
25
30
PC
PMMA
I
(a) (b)
Time (s)
Amplitude of Harmonic Displacement (nm)
0 50 100 150 200 250 300
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
PC
PMMA
Time (s)
Outofphase angle (Degree)
0 50 100 150 200
0
5
10
15
20
25
30
PC
PMMA
(c) (d)
Fig. 47 Nanoindentation output from oscillation on a step loading at 75 Hz. (a) Carrier
loaddepth curve. (b) The history of harmonic load amplitude. (c) Response of harmonic
displacement amplitude. (d) Outofphase angle with correction.
In the context of presentation of complex compliance in this work, Eq. (410) is used at
all frequencies under a harmonic loading superimposed on a step loading, but results in
nanoindentation at frequencies higher than c
should be understood as a very good
51
approximation to the actual values due to the high correlation between approximate and
accurate solutions.
The nanoindentation results of input and output under the second dynamic loading at
75 Hz are shown in Fig. 47. Fig. 47(a) records the corresponding quasistatic carrier
loaddepth curve (without the harmonic component) for both PC and PMMA. Figs. 47(b)
and (c) illustrate the amplitudes of harmonic load and displacement at 75 Hz for PC and
PMMA. As shown in Fig. 47(b), at initial stage of the contact between the indenter and
sample surface, the amplitude of harmonic load generally increased until t = 50 s, and
was maintained constant thereafter. Accordingly, as shown in Fig. 47(c), there was an
increase in amplitude of harmonic displacement before t = 50 s. So the condition of Eq.
(44) is precise after about 50 s. A steady state was reached at about t = 150 s. As shown
in Fig. 47, after t = 150 s, m h / h 0  is much less than 1%, thus the condition of using Eq.
(410) is satisfied. Fig. 47(d) shows the outofphase angle between harmonic load and
harmonic displacement for both PC and PMMA.
In order to ensure that the deformation of the polymer samples is in the linearly
viscoelastic regime, the indentation depth into the sample surface for PC and PMMA
materials was controlled to within the limit of linearity. According to Lu, et al, (2003), the
limits of linearity in indentation depth were determined as 1123 nm for PC, and 780 nm
for PMMA, respectively, for a spherical indenter with a radius of 3.4 μm. It was found
that in indentation within the limit of linearity, the deformation of PC and PMMA is
linearly viscoelastic, indicated by the fact that complex compliance is independent of the
magnitude of indentation carrier load.
52
For timedependent materials under dynamic loading, the outofphase angle between
the harmonic load and displacement plays an important role in the computation of
complex compliance. Consequently, we discuss next the correction on outofphase angle.
As presented by Pethica and Oliver (1992), the outofphase angle between the force
and displacement 7 can be computed by
2
( )
tan
7
k m
C Ci c
−
+
= , (418)
where k is system stiffness, c C is the damping inside capacitor gauge measuring
displacement of the indenter, and i C is the damping resulting from the contact between
indenter and sample. If the sample material is ideally elastic, there is no outofphase
angle between harmonic indention load and harmonic indentation depth. For an elastic
material, however, from Eq. (418), = 0 i C , but 8 0, c C so that 7 8 0 , which is not
physically reasonable. It should be noted that the air damping resulting from the gauge
capacitor gives apparent outofphase angle that does not necessarily represent the
damping behavior of the material. Therefore, 7 is not exactly the outofphase angle
between harmonic indentation load and harmonic depth and must be corrected. If the
contribution in 7 from the nanoindentation instrument is removed, we can determine the
outofphase angle, / , representing the damping of viscoelastic material by
2 tan
/
k m
Ci
−
= . (419)
As an example, the outofphase angles for PC and PMMA using spherical indenter are
shown in Fig. 45(d) and Fig. 47(d).
53
Log (Frequency) (Hz)
Complex Compliance in Shear (1/GPa)
0 1 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
DMA (PC)
DMA (PMMA)
Nanoindentation (PC)
Nanoindentation (PMMA)
Storage Compliance in Shear
Loss Compliance in Shear
Fig. 48 Complex compliance in shear from nanoindentation under a harmonic load
superimposed on a constant rate loading.
To determine the complex compliance from nanoindentation tests using the spherical
indenter, Eq. (410) is used. The complex compliance curves of PC and PMMA materials
are shown in Figs. 48 and 49. Error bars are shown for storage compliance. Error bars
for loss compliance are about 1/3 size of the symbols. The complex compliance shown in
Fig. 48 was measured from nanoindentation under a constant rate loading superimposed
by a harmonic load. Fig. 49 shows the complex compliance measured from
nanoindentation under a step loading superimposed by a harmonic load. The complex
compliance measured from both types of nanoindentation tests were compared with
DMA results. These two sets of results are in good agreement. The average percent error
for the storage compliance of PC and PMMA at these discrete experimental data is less
than 6%, indicating a very good agreement. The maximum errors for the storage
compliance of PC and PMMA are 9.1% and 5.1%, respectively. Figs. 48 and 49 show
that the loss compliances for both PC and PMMA are much smaller than the storage
compliances, which implies that the material damping of the two polymers is very small.
54
For PC, both storage and loss compliances remain almost constant within 3  200 Hz. The
reductions in storage and loss compliance for PC are only 0.8% and 3.8%, respectively,
while for PMMA, the storage compliance decreases by 16.6%, and loss compliance
decreases by 35.4% between 3 Hz and 260 Hz.
Log (Frequency) (Hz)
Complex Compliance in Shear (1/GPa)
0 1 2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
DMA (PC)
DMA (PMMA)
Nanoindentation (PC)
Nanoindentation (PMMA)
Storage Compliance in Shear
Loss Compliance in Shear
Fig. 49 Complex compliance in shear from nanoindentation under a harmonic load
superimposed on a step loading.
In the computation of both storage compliance and loss compliance, Poisson’s ratio is
usually timedependent for polymeric materials. In these experiments, the timescale is
not large; the maximum time duration is less than 300 seconds. Within this time period, it
has been demonstrated that the Poisson’s ratio does not change significantly (Lu, et al,
1997). Therefore, we assumed a constant Poisson’s ratio ($ = 0.3 ) in the computation of
the complex compliance for both PC and PMMA materials. The natural frequency of the
indentation system is 180 Hz, thus we avoided testing near the resonant frequency.
We turn next to the comparison in results from Loubet et al. and the new method
proposed in this study. The method by Loubet et al. (1995) on measurement of the
55
uniaxial complex modulus is based on the analogy between dynamic indentation and
uniaxial dynamic analysis. Nanoindentation on viscoelastic materials, however, is a
complex viscoelastic problem involving moving contact interface, and needs to be
analyzed to derive formulas for the computation of viscoelastic functions in the
frequencydomain. Based on their method, the uniaxial storage modulus at 75 Hz for both
PC and PMMA were computed using the data of stiffness, damping and contact area
from nanoindentation experiments, as shown in Fig. 41. During the tests, a constant rate
loading superimposed by a small harmonic loading was applied. Since tests were under a
single frequency (75 Hz) within a time frame less than 250 s, the storage modulus
computed using Eq. (41) was plotted with respect to time. Also plotted in Fig. 41 are
the DMA data of uniaxial storage modulus at 75Hz, which is a horizontal line since it is a
single value representing material behavior. The uniaxial storage modulus measured from
this method is generally much higher than DMA data. As shown in Fig. 41, the
minimum errors in storage modulus are 40% for PC and 46% for PMMA, respectively,
indicating that the approach based on the analogy between uniaxial dynamic tension test
and dynamic nanoindentation is not appropriate for measuring viscoelastic functions for
polymers. Also plotted in Fig. 41 are the storage modulus data using the new method
presented. Eq. (413) was used to convert shear data to uniaxial data. For PC in steady
state, the average storage modulus obtained from the new method is consistent with the
DMA data, which is 2.29 GPa, and the error is 6.1%. For the average storage modulus of
PMMA in steady state, the error, compared with the conventional data from DMA, is
2.5%. The indentation results using the new method for timedependent materials, such
as polymers, can recover complex viscoelastic functions determined from conventional
56
tests with good accuracy. The complex compliance in shear measured using the proposed
method at all frequencies shows a good agreement with those obtained from conventional
tests. This study has thus provided a method to measure viscoelastic functions in the
frequencydomain for timedependent materials.
4.5 Conclusions
A method to measure the complex compliance has been presented using
nanoindentation with a spherical indenter for linearly viscoelastic materials. Following
the Hertzian solutions for indentation in linear elasticity and the consideration of the Lee
Radok approach for a moving boundary problem in linear viscoelasticity, formulas for
the components of the complex compliance function in the frequencydomain have been
derived based on the loaddisplacement relation for linearly viscoelastic materials under
harmonic loading. The formulas should be used under the conditions that h/R < 0.16 and
the nanoindentation depth is below the limit of linearity. When a constant rate loading is
used as the carrier load, the formulas are exact under the condition that the loading rate is
high enough so that the condition of nondecreasing contact area is satisfied. While a step
loading is used as the carrier load, the formulas are considered to be approximate when
the frequency is higher than a critical value such that decreasing contact area occurs; in
the case of decreasing contact area, the Ting approach was used to find the solution in the
steady state at a selected frequency to estimate the difference between solutions obtained
using the LeeRadok approach and the Ting approach. Very close correlation was found
for the test conditions used in this study. Dynamic nanoindentation tests on PC and
PMMA materials were performed, using a spherical indenter, to determine the amplitudes
57
of oscillating load and displacement, as well as the outofphase angle of displacement
with respect to the harmonic force output. The complex compliance functions in the
frequencydomain were determined using the proposed method and results were
compared with conventional data obtained from DMA tests for PC and PMMA materials.
The condition for nondecreasing contact area, 0 0
v P , is satisfied under a constant
rate carrier load so that the complex compliance formulas derived from the LeeRadok
approach are appropriate. Under a step carrier load, the condition of nondecreasing
contact is satisfied up to the frequency limit, 0 h h m &  ; at frequencies higher than the
frequency limit, complex compliance data could be considered to be an approximation. In
both constant rate and step carrier loading conditions, a good agreement between
nanoindentation results and conventional data has been reached, indicating the validity of
the proposed method for measuring the complex compliance function in the frequency
domain using dynamic nanoindentation.
58
CHAPTER V
MEASUREMENTS OF TWO INDEPENDENT VISCOELASTIC
FUNCTIONS BY NANOINDENTATION
5.1 Introduction
In measurements of elastic properties, Poisson’s ratio is often of interest; however
it is hardly thoroughly studied. Lucas, Hay and Oliver (2004) applied a lateral load to
determine the Poisson’s ratio through the measurements and analysis of both normal
contact stiffness and tangential contact stiffness. Various methods have been developed
to measure the Poisson’s ratio for a linearly viscoelastic material at the macroscale.
Vlassak and Nix (1992) used bulge tests on both square and rectangular membranes, and
measured both Young’s modulus and Poisson’s ratio for silicon nitride; Ma and Ravi
Chandar (2000) used a cylinder polymer sample under confined compression to measure
both bulk and shear relaxation functions. These techniques are successful in their
respective areas of application.
To date, methods are not available to measure two independent viscoelastic functions,
such as bulk and shear relaxation functions at micro/nano scale using nanoindentation. In
all the current nanoindentation techniques for measurements of viscoelastic functions, a
constant Poisson’s ratio is often assumed, and nanoindentation measures only one
viscoelastic function(Cheng et al., 2000; Lu et al., 2003; Huang et al., 2004; Odegard et
59
al., 2005; VanLandingham et al.; 2005, Cheng and Cheng, 2005), such as the creep
compliance in shear. However, for very small amounts of materials with viscoelastic
behavior different from a bulk material, all viscoelastic functions are unknown. In
addition, for a material with pronounced viscoelastic effects, such as a polymer near its
glass transition temperature, any pair of independent viscoelastic functions (e.g., bulk and
shear relaxation functions) would change with time. Assuming a constant Poisson’s ratio
could potentially cause significant error. Consequently, a method is needed to measure
two independent viscoelastic functions using nanoindentation for very small amounts of
viscoelastic materials.
In this chapter, a method is presented to measure two independent viscoelastic
functions, namely bulk and shear relaxation functions for an isotropic, linearly
viscoelastic material using nanoindentation. Equations are derived for the viscoelastic
contact mechanics problems for both Berkovich and spherical indenters. The bulk and
shear relaxation functions are determined through minimizing the difference between
nanoindentation data and analytical results. The results from nanoindentation will be
compared with viscoelastic property data determined from conventional tests for the
same batch of materials to examine the method.
5.2 Analytical background
In this section we present formulas for the indentation loaddisplacement relation
from linearly elastic contact mechanics analysis, and then use the approach developed by
LeeRadok (1960) to write down the indentation loaddisplacement relation for a linearly
viscoelastic material.
60
5.2.1 Indentation by rigid axisymmetric indenters of arbitrary shape
For the Boussinesq problem of a rigid axisymmetric indenter tip indenting into a halfspace
composed of a homogeneous, linearly elastic and isotropic material, Sneddon
(1965) derived solution for the indentation load P given as
− −
=
1
0
2
2 '
1
( )
1
4
dx
x
a x f x
P
$
μ
, (51)
where μ is the shear modulus, $ is the Poisson’s ratio, a is the contact radius and
z = f (x) is the shape function of the indenter with x = r / a as defined in Fig. 51.
Sneddon’s solution for the indentation displacement h is
−
=
1
0
2
'
1
( )
dx
x
f x
h . (52)
Since x = r / a , Eq. (52) gives the relationship between displacement of the axisymmetric
indenter tip, h, and the contact radius, a. The relationship between h and a is uniquely
determined by the geometry of the indenter. Combining Eqs. (51) and (52), and
expressing $ in terms of bulk modulus, K and shear modulus, μ , one has
( )
3 4
(3 )
F h
K
K
P
μ
μ μ
+
+
= , (53)
where F(h) is a function determined by Eqs. (51) and (52), and depends on the
geometry of the axisymmetric indenter. For example, for a conical indenter,
( ) 8 /( tan ) F h = h2 , where is the angle between the cone generator and the surface
of the half space.
61
Fig. 51 A schematic of indentation on a half space by an axisymmetric indenter
When a rigid axisymmetric indenter indents into a half space composed of a
homogeneous, isotropic and linearly viscoelastic material, following the approach by
LeeRadok (1960), under the condition of nondecreasing contact area between the
indenter and the workpiece, the loaddisplacement relation in the Laplace domain can be
written in the form
( )
( ( ))
( )[3 ( ) ( )]
3 ( ) 4 ( )
P s
F h s
s s K s s
K s s =
+
+
μ μ
μ
, (54)
where s is the complex variable in Laplace domain, the notation Q(s) represents the
Laplace transform of function Q(t), for example, K , μ , are the Laplace transform of
bulk relaxation function K(t) and shear relation function μ(t), respectively. It is noted that
K(s) and μ ( s) can only be determined to the extent as shown on the left hand side of Eq.
(54) when an axisymmetric nanoindenter is used, the two functions cannot be separated
further. In other words, both K and μ can only be determined in terms of their ratio in
Laplace domain when any combinations of axisymmetric indenters are used. For example,
Indenter
Sample
2a
x = r / a Z = f (x)
2r
62
for a conical indenter that has the shape function, f (x) = ax tan , the following relation
in the Laplace domain holds
μ μ
μ
( ) tan
8 ( )
( )[3 ( ) ( )]
3 ( ) 4 ( ) 2
P s
h s
s s K s s
K s s =
+
+
. (55)
For a circular flat punch indenter, the following relation holds,
( )
8 ( )
( )[3 ( ) ( )]
3 ( ) 4 ( )
P s
Rh s
s s K s s
K s s =
+
+
μ μ
μ
, (56)
and for a spherical indenter, the shape function is ( ) /(2 ) 2 2 f x = a x R under condition
ax<<R, the following relation in the Laplace domain holds
3 ( )
16 ( )
( )[3 ( ) ( )]
3 ( ) 4 ( ) 3 / 2
P s
Rh s
s s K s s
K s s =
+
+
μ μ
μ
. (57)
It is seen on the left hand sides of Eqs. (55)  (57) that, with the use of one or more
equations in Eqs. (55)  (57), one can only determine
( )[3 ( ) ( )]
3 ( ) 4 ( )
s s K s s
K s s
μ μ
μ
+
+
, and cannot
separate K( s ) from μ( s ) . Consequently, using any two different axisymmetric indenters
cannot determine two independent viscoelastic functions. To separate and determine the
two independent functions, we need another independent asymmetric nanoindentation
problem.
63
hc hc
(a) Berkovich and conical indenters
R
h
a
x=r/a
Z=f (x)
(b) Spherical indenter
Fig. 52 Berkovich, conical and spherical indenters
5.2.2 Berkovich indenter
Berkovich indenter is usually used in nanoindentation due to primarily its selfsimilarity
in geometry. Berkovich indenter is often modeled as a conical indenter based
on approximately the same heightarea relationship of the two indenter tips. However, the
Berkovich indenter has a flipped threeface pyramidal shape, as shown in Fig. 52, and is
64
not axisymmetric. Consequently Berkovich indenter is not a rigorously axisymmetric
indenter, and represents some difference from an axisymmetric conical indenter.
Since the nanoindentation by Berkovich indenter is not an axisymmetric problem, it is
necessary to determine the loaddisplacement relation for indentation by Berkovich
indenter to establish a second equation independent of Eq. (54) (with special cases given
in Eqs. (55)  (57)). However, an analytical solution for the viscoelastic contact problem
using an asymmetric Berkovich indenter is not available. Consequently, a semiempirical
approach is taken to determine the loaddisplacement relation when the material is
considered linearly viscoelastic. To this end, the nanoindentation by a Berkovich indenter
on a linearly elastic material is modeled first, and then the linearly elastic solution is
extended to linearly viscoelastic solution. To determine a semiempirical linearly elastic
solution, the indentation by Berkovich indenter tip was simulated using
ABAQUS/Standard (2004) on different elastic materials with a series of Poisson’s ratios
and fixed Young’s modulus. The simulations were also conducted on elastic materials
with different Young’s moduli and fixed Poisson’s ratio, and it was found that the load P
is proportional to Young’s modulus or shear modulus at the same indentation
displacement. Therefore it is possible to fit the numerical results on the relation between
P and h into an analytic representation. The loaddisplacement curves at some Poisson’s
ratios with a fixed Young’s modulus are shown in Fig. 53(a), and the curves were fitted
numerically, to the following equation
2
1
(1 0.2202 )
2.0837 h
v
P
−
−
=
$ μ
. (58)
65
Displacement (nm)
Load (mN)
0 200 400 600 800 1000
0
1
2
3
4 0.4
0.37
0.3
0.25
0.2
$ = 0.1
$ =
$ =
$ =
$ =
$ =
(a) FEM results of loaddisplacement curves at different Poisson’s ratios
Displacement (nm)
Load (mN)
0 200 400 600 800 1000
0
1
2
3
4
$=0.4
$=0.1
Fitting
(b) Curve fitting of loaddisplacement data from FEM
Fig. 53 FEM modeling
As an example, both finite element results and fitted curves, indicated by Eq. (58), are
plotted in Fig. 53(b). The two sets of results are very close to each other, with the
minimum value of crosscorrelation coefficient being 0.99992 for all Poisson’s ratios. It
66
is noted that in the parenthesis of Eq. (58), only linear term of Poisson’s ratio is used for
purpose of simplifying the viscoelastic analysis in the sequel. Nonlinear terms in the
parenthesis of Eq. (58) can provide slightly higher accuracy, but will lead to higher level
of complexity in viscoelastic analysis, and are not used in this analysis.
5.2.3 Viscoelastic solutions
Eq. (58) shows that Berkovich indenter has a loaddisplacement relationship
different from a conical indenter which is
2
tan 1
4
P h
$
μ
−
= . (59)
The representations in terms of elastic parameters are different in equations for
indentations by Berkovich and spherical indenters. The situation is similar in the
solutions to viscoelastic indentations by two indenters using hereditary integral operators
(Lee and Radok, 1960). Consequently, using both Berkovich indenter and an
axisymmetric indenter would potentially lead to the measurements of two independent
viscoelastic functions. In this study, a spherical indenter is used, however, the similar
approach can be used for a conical indenter or a circular flat punch indenter.
The Hertzian solution (Hertz, 1881) for indentation loaddisplacement relation by a
spherical indenter indenting into a homogeneous, isotropic and linearly elastic material is
3 / 2
3(1 )
8
h
v
R
P μ
−
= . (510)
67
Eq. (58) can be rewritten in terms of bulk modulus K and shear modulusμ for the
Berkovich indenter as
2
1
2
(3K + 4μ )P1 = 8.3348(2.6694Kμ +1.2202μ )h , (511)
where P1 and h1 are the indentation load and displacement, respectively, under
indentation by a Berkovich indenter. For indentation by a spherical indenter, Eq. (512)
can be rewritten as
3 / 2
2
2
2 (6 2 )
3
8
(3 4 ) K h
R
K + μ P = μ + μ , (512)
where P2 and h2 are the indentation load and displacement, respectively, under
indentation by a spherical indenter. It is noted that viscoelastic indentations using either a
Berkovich indenter or a spherical indenter involves varying contact area between an
indenter and the work material. Consequently, the correspondence principle, requiring a
fixed (i.e., timeindependent) displacement boundary, cannot be applied directly. For the
timevarying displacement boundary problem as in nanoindentation of a linearly
viscoelastic material, Lee and Rodok (1960) developed an approach to use the hereditary
integral operators to determine the relation between indentation load and displacement.
Using the LeeRadok approach for indentations by Berkovich and spherical indenters the
loaddisplacement relations are
,
( )
10.1718 ( ) ( )
( )
22.2489 ( ) ( )
( )
4 ( )
( )
3 ( )
0
2
1
0
0
2
1
0 0
1
0
1
μ
μ
μ
μ
d d
d
dh
d
d
t
d d
d
dh
K
d
d
d t
d
dP
d t
d
dP
K t
t
t t t
+ − −
− + − = − −
(513)
68
.
( )
( ) ( )
3
16
( )
16 ( ) ( )
( )
4 ( )
( )
3 ( )
0
3 / 2
2
0
0
3 / 2
2
0 0
2
0
2
μ
μ
μ
μ
d d
d
dh
d
d
t
R
d d
d
dh
K
d
d
d R t
d
dP
d t
d
dP
K t
t
t t t
+ − −
− + − = − −
(514)
It is noted that a nondecreasing contact area between the indenter and the workpiece
should be maintained for Eqs. (513) and (514) to be valid.
Under constant rate loading histories, P V t 1 1 = and P V t 2 2 = , with 1 V and 2 V being
constant loading rates, Eqs. (513) and (514) are simplified to
[ − + − ] = − +
t t
d V J t K d V t
d
dh
K t t
0
1 1
2
1
0
3 ( ) ( ) 4
( )
22.2489 ( ) 10.1718 ( )
μ , (515)
[ − + − ] = − +
t t
d V J t K d V t
d
dh
K t t
R
0
2 2
3 / 2
2
0
3 ( ) ( ) 4
( )
3 ( ) ( )
3
16
μ , (516)
where J(t) is creep compliance in shear.
Under a constant rate loading history, both h1(t) and h2(t) can be measured from
nanoindentation. Solving Eqs. (515) and (516) should lead to the determination of two
independent viscoelastic functions K(t) and μ (t) . However, Eqs. (515) and (516) are
difficult to solve directly. To circumvent this difficulty, a least squares correlation
approach between experimentally measured displacements h1(t) and h2(t) and the
corresponding analytical values is taken and described as follows.
The bulk and shear relaxation functions can be represented by the generalized
Maxwell model:
69
+=
−
= +
N
i
t
i
K t K K e i
1
/ ( ) # and +=
−
= +
N
i
t
i
t e i
1
/ ( ) μ μ μ # , (517)
where i μ
and i K are relaxation numbers, i #
the relaxation times and N the number of
exponential terms in the Prony series.
Define a least squares correlation coefficient, C,
( )
( )
( )
( )2
1
exp
2,
2
1
2,
exp
2,
2
1
exp
1,
2
1
1,
exp
1,
+
+
+
+
=
=
=
=
−
+
−
=
M
i
i
M
i
th
i i
M
i
i
M
i
th
i i
h
h h
h
h h
C , (518)
where exp
1,i h and exp
2,i h are the measured displacement data at time i t for Berkovich
indentation and spherical indentation, respectively; th
i h 1, and th
i h 2, are analytical results of
displacements computed from Eqs. (515) and (516), respectively, and can be
represented in terms of the parameters in Eq. (517) after equations in Eq. (517) are
substituted into Eqs. (515) and (516). When the measured displacements can be fully
described by displacements computed by Eqs. (515) and (516) using appropriate
parameters in Eq. (517), ideally C would be zero. In reality, however, C will not reach
zero; instead, it must be minimized with the use of appropriate parameters in Eq. (517).
During the minimization process, the bestfit parameters in Eq. (517) are iteratively
searched until the coefficient C is minimized. The minimization of C will converge when
two minimizations, one for the Berkovich indentation and the other for the spherical
indentation, as shown in Eq. (518) are simultaneously achieved, indicated by the best
correlations between nanoindentation loaddisplacement curves determined from both
nanoindentation data and analytical results for each indenter. Thus, minimizing C leads to
70
a set of appropriate parameters in Eq. (517) for determining two independent
viscoelastic functions, K(t) and μ (t) .
5.3 Nanoindentation measurements
An MTS Nano Indenter XP system was used in nanoindentation tests to acquire loaddisplacement
data. The nanoindenter can reach a maximum indentation depth of 500 μm
and a maximum load of 500 mN. The displacement resolution is 0.2 nm and the load
resolution is 50 nN. Both the Berkovich and spherical indenters are made of diamond;
their schematic geometries are as shown in Fig. 52. The Berkovich indenter has a threefaced
pyramidal tip, and the spherical indenter has a tip radius of 10 μm. In all
nanoindentation with the spherical indenter, the maximum indentation depth was below
620 nm.
The materials used in these tests were poly(vinyl acetate) (PVAc) and poly(methyl
methacrylate) PMMA. The PVAc resin was the same as used in the work by Knauss and
Kenner (1980), and by Deng and Knauss (1997); the resin was stored in an airtight
container, and was molded using the same procedures as in their work. The PMMA
samples were made from the same PMMA plate as used in the work by Lu, et al. (1997)
and by Sane and Knauss (2001). The PVAc specimen has a glass transition temperature
of 29 ºC and the PMMA specimen has a glass transition temperature of 105 ºC. The
dimensions of PVAc and PMMA specimens were 20mm×20mm×6mm and
20mm×10mm×5mm, respectively. The PVAc specimen was annealed at 34 ºC and
PMMA specimen was annealed at 110 ºC for two hours, and they were cooled down
71
slowly to room temperature at a cooling rate of approximately 5 ºC /hr. Samples were
then stored in an enclosed desiccator with approximately 50% relative humidity produced
by placing a saturated salt solution in this enclosed environment. The specimens were
then carefully mounted on aluminum holders. All specimens had ageing time of nearly 75
hours. The humidity in the room was maintained at ~ 50% relative humidity.
The nanoindentation tests on PVAc were performed in air at 30 ºC. An infrared bulb
placed close to the floor inside the nanoindenter chamber was used to heat the enclosed
nanoindentation system to the desired temperature. The temperature was monitored and
controlled by a temperature controller (Chromalox Instruments and Controls, Model 1604)
with a resolution of ±0.1 ºC. Proper thermal insulation was used on the MTS Nano
Indenter XP system to maintain the temperature stability during nanoindentation. The
nanoindenter was calibrated with the use of a fused silica sample to ensure that loaddisplacement
outputs were accurate at 30 ºC. The nanoindentation tests on PMMA were
conducted at room temperature (23 ºC). Each test did not start until the drift rate of the
indenter tip had dropped below a set value (typically 0.05 nm/s) to ensure that a thermal
equilibrium condition for the specimen and nanoindenter system had been reached. This
procedure is necessary as the precision of the nanoindenter depends on the temperature
gradient of the instrumentation. After the indenter tip had made contact with the
specimen surface, a constant rate indentation load was applied, and both the indentation
load and indentation depth were recorded simultaneously at a sampling rate of five data
points per second.
72
5.4 Results and discussions
Results on bulk and shear relaxation moduli are reported and discussed in this section
from nanoindentation measurements using both Berkovich and spherical indenter tips.
Constant rate loading histories were used in all nanoindentation tests.
For PVAc, nanoindentation tests were carried out at 30 ºC, right in the glass transition
region (Tg = 29 ºC). A constant rate loading at a loading rate of 19.8 μN/s was applied for
both Berkovich and spherical indenters. The entire nanoindentation duration was less
than 120 s in order to ensure thermal stability during each test. The loaddisplacement
curves for PVAc from nanoindentation using both Berkovich and spherical tips are
shown in Fig. 54. Data scattering from different tests are indicated by the error bars. As
shown in Fig. 54, the repeatability of loaddisplacement data was high.
Displacement (nm)
Load (mN)
0 200 400 600 800 1000
0
0.5
1
1.5
2
2.5
Berkovich nanoindentation tests
Spherical nanoindentation tests
Fig. 54 Nanoindentation loaddisplacement curves for PVAc
73
The loaddisplacement data obtained from nanoindentation were analyzed using Eqs. (5
15)  (518) to determine both bulk and shear relaxation functions. An iterative algorithm
was applied to extract two independent relaxation functions from minimization of the
least squares coefficient C in Eq. (518). The bulk and shear relaxation functions were
determined when the best correlation between loaddisplacement curves from
nanoindentation and analysis was achieved. Both the measured and analytical loaddisplacement
curves are plotted in Fig. 55(a) and Fig. 55(b) for indentations by
Berkovich indenter and spherical indenter, respectively. The cross correlation
coefficients between the two loaddisplacement curves for Berkovich indenter and
spherical indenter are 0.9998 and 0.9999, respectively. The good correlations were
reached simultaneously in the two sets of Ph curves for both indenters, indicating the
convergence of the overall minimization of C. The minimum C as computed from Eq. (5
18) is 0.000992.
Displacement (nm)
Load (mN)
200 400 600 800
0
0.5
1
1.5
2
2.5
Experiment
Analytical
(a) Berkovich indenter
74
Displacement (nm)
Load (mN)
50 100 150 200 250 300
0
0.5
1
1.5
2
Experiment
Analytical
(b)Spherical indenter
Fig. 55 Minimization results of loaddisplacement curves for PVAc
Minimizing C leads to the following bulk and shear relaxation functions for PVAc
t t K t e e 0.05 0.1 ( ) 2.801 0.448 0.252 = + − + − GPa, (519)
t t t e e 0.05 0.1 ( ) 1.102 0.331 0.165 μ = + − + − GPa. (520)
Time (s)
Bulk and Shear Modulus (GPa)
20 40 60 80 100
0
1
2
3
4
Conventional
Nanoindentation
Bulk Relaxation Modulus
Shear Relaxation Modulus
(Knauss and Kenner, 1980)
(Deng and Knauss, 1997)
Fig. 56 Results of K(t) and μ (t) for PVAc from nanoindentation
75
The bulk and shear relaxation functions given by Eqs. (519) and (520) are shown in Fig.
56. Also plotted in Fig. 56 are the conventional data. The conventional bulk relaxation
modulus for PVAc are converted from complex bulk compliance data obtained by Deng
and Knauss (1997), and the shear relaxation modulus data were measured by Knauss and
Kenner (1980) on PVAc samples made of the same PVAc resin following the same
molding procedures. An approximation method for data conversion as described by Emri
et al. (2005) was used to convert complex bulk modulus (reciprocal of complex bulk
compliance) in the frequency domain to bulk relaxation modulus in the time domain. As
shown in Fig. 56, the bulk and shear relaxation functions are in a reasonably good
agreement with data obtained from conventional tests. The average errors for bulk and
shear relaxation moduli are 9.54% and 2.64%, respectively. The following sources could
contribute to the discrepancy in the data from nanoindentation in this work and from
conventional tests: (1) the conventional bulk relaxation modulus was not measured
directly in the time domain, and was converted from complex bulk modulus in the
frequency domain. Using an approximate method to convert data in the frequency
domain to the time domain could cause error; and (2) nonlinear deformation in a small
area in the workpiece close to indenter was not considered in the model.
76
Displacement (nm)
Load (mN)
0 200 400 600 800 1000 1200
0
1
2
3
4
5
6
7
8
Berkovich nanoindentation tests
Spherical nanoindentation tests
Fig. 57 Nanoindentation loaddisplacement curves for PMMA
Nanoindentation on PMMA was conducted at 23 ºC. A constant rate loading at a loading
rate 33.0 8N/s was used for nanoindentation using both Berkovich and spherical
indenters. The loaddisplacement curves from nanoindentation measurements are shown
in Fig. 57; the results were very reproducible for each indenter, indicated by the error
bars showing data scattering. The loaddisplacement data were analyzed using Eqs. (5
15)  (518) to minimize the least squares correlation coefficient defined in Eq. (518) so
that a best set of parameters in the Prony series in Eq. (517) can be determined to allow
the analytical loaddisplacement data to correlate with data from nanoindentation
measurements. The best set of parameters in the Prony series are then used in Eq. (517)
to determine the two independent viscoelastic functions, bulk and shear relaxation
functions. The results on nanoindentation loaddisplacement data for the use of
Berkovich and spherical indenters are shown in Fig. 58(a) and Fig. 58(b), respectively.
The cross correlation coefficient between analytical results and the data from
77
nanoindentation measurements is 0.9991 for indentation by a Berkovich indenter, and is
0.9997 for indentation by a spherical indenter. The good correlation for both indenters, as
shown in Fig. 58, leads to the minimization of C, which is 0.000699, as computed from
Eq. (518).
Displacement (nm)
Load (mN)
0 200 400 600 800
0
1
2
3
4
Experiment
Analytical
(a) Berkovich indenter
Displacement (nm)
Load (mN)
0 50 100 150 200 250 300 350
0
1
2
3
Experiment
Analytical
(b) Spherical indenter
Fig. 58 Minimization results of loaddisplacement curves for PMMA
78
Results of the bulk and shear relaxation functions for PMMA as determined from this
approach are
t t K t e e 0.01 0.1 ( ) 4.800 0.336 0.288 = + − + − GPa , (521)
t t t e e 0.05 0.1 ( ) 1.001 0.075 0.060 μ = + − + − GPa. (522)
The bulk and shear relaxation functions are shown in Fig. 59, and are compared with
conventional data. The conventional data of bulk relaxation modulus were converted
from the complex bulk compliance measured by Sane and Knauss (2001), and the
conventional shear relaxation data were measured by Lu, Zhang and Knauss (1997). As
shown in Fig. 59, the data measured using the nanoindentation agree reasonably well
with conventional data. The average errors for the measured bulk and shear relaxation
moduli are 4.8% and 5.9%, respectively.
Since the temperature for nanoindentation on PVAc was 30 ºC, slightly higher than
its glass transition temperature (Tg = 29 ºC), the PVAc was in the glass transition region,
so that relaxation was significant. For example, as shown in Fig. 56 the measured shear
relaxation modulus decreased by 30.9% for PVAc within 100 s from the beginning of
tests, representing a very pronounced viscoelastic behavior. For PMMA, nanoindentation
was conducted at 23 ºC, much lower than its glass transition temperature (Tg = 105 ºC).
The shear relaxation modulus decreased by 11.9% within 100 s, as shown in Fig. 59,
indicating that the relaxation behavior of PMMA in the glassy state is present, but not as
significant as that of PVAc in the glass transition region.
79
Time (s)
Shear or Bulk Relaxation Modulus (GPa)
50 100 150
0
1
2
3
4
5
Conventional
Nanoindentation
Bulk Modulus
Shear Modulus
(Sane and Knauss, 2001)
(Lu, et. al., 1997)
Fig. 59 Results of K(t) and μ (t) for PMMA from nanoindentation
It is noted that for nanoindentation using spherical indenter, the Hertzian solution
holds only when the ratio of indentation depth to the radius of spherical indenter is small,
for example, less than ~ 0.16 (Giannakopoulos, 2000). For all spherical indentations in
this study, the maximum indentation depth used was less than 350 nm, so that h < 0.05R ,
thus the use of Herzian solution was justified.
In order to ensure that the deformation of the polymer samples is in the linearly
viscoelastic regime, the maximum indentation depth into the sample surface for PMMA
and PVAc materials was controlled to within the limit of linearity. According to Lu, et al.
(2003), the limit of linearity in indentation depth under constant rate loading condition
was determined as 780 nm for PMMA for Berkovich indentation. It was found that in
indentation within the limit of linearity, the deformation of PMMA is (nearly) linearly
viscoelastic. For PVAc, since the specimen was in the rubbery state (above glass
transition temperature) in nanoindentation, the limit of linearity was considered to be
80
higher than that of glassy state (below glass transition temperature). Consequently, by the
use of a maximum indentation displacement of ~ 800 nm for PVAc, the PVAc would stay
within limit of linearity.
Measurements of two independent viscoelastic functions are a wellknown
challenging problem, even at macroscale, due to excessive accuracy needed to acquire
two independent sets of timedependent data (Lu et al., 1997; Tschoegl et al., 2002) The
method presented here has potential for measurements of two independent viscoelastic
functions at submicron scale. Recently the commercially available nanoindenter can
reach a temperature range between 10 ºC and 200 ºC. The method presented here can be
applied to this range of temperatures so that both bulk and shear relaxation functions can
be determined over a wide range of temperatures to form master curves as necessary
information for the prediction of longterm viscoelastic behavior necessary for the
investigation of longterm durability of very small amounts of timedependent materials
such as coatings in medical devices, MEMS and NEMS.
5.5 Conclusions
A method has been developed to extract two independent viscoelastic functions from
nanoindentation data. Based on the difference of representations of two independent
viscoelastic functions in the loaddisplacement