2

Materials Science and Engineering A 385 (2004) 74–82

A simple phenomenological approach to nanoindentation creep
A.C. Fischer-Cripps∗
CSIRO Division of Telecommunications and Industrial Physics, P.O. Box 218, Lindfield, NSW 2070, Australia

Received 5 February 2004; received in revised form 19 April 2004

Abstract

Nanoindentation is frequently used to measure elastic modulus and hardness of structural materials such as ceramics, metals and thin films.
The assumption behind conventional nanoindentation analysis methods, where the unloading data is analysed, is that the material behaves in an
elastic-plastic manner. However, many materials can also exhibit a visco-elastic and visco-plastic response which is commonly termed “creep”.
In a nanoindentation test, this is usually observed as an increase in depth during a hold period at maximum load in the load-displacement data.
Creep is not accommodated in conventional nanoindentation analysis methods. The present work shows how conventional linear spring and
dashpot elements can be used to model the creep response of a wide range of materials using the hold period force-displacement data. The
method shown can be readily incorporated into a computer program and can be used with any conventional nanoindentation test instrument
using either spherical or sharp indenters.
Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved.

Keywords: Nanoindentation; Indentation creep; Depth-sensing indentation

1. Introduction properties of the specimen material. It is the second method
that is the subject of the present work.
Nanoindentation has proven to be an effective and con- In a nanoindentation test, creep and plastic deforma-
venient method of determining the mechanical properties tion in the conventional sense, i.e., that occurring due to
of solids, most notably elastic modulus and hardness. The shear-driven slip for example, should be regarded separately.
most popular method relies on an analysis of the unloading Plasticity, in the sense of yield or hardness, is conveniently
load-displacement response which is assumed to be elastic, thought of as being an instantaneous event (although in
even if the contact is elastic-plastic [1]. The method relies practice, it can take time for yield processes to complete).
on plasticity occurring instantaneously upon satisfaction of In contrast, creep can occur over time in an otherwise elas-
a constitutive criterion and that there are no time-dependent tic deformation as a result of the diffusion and motion of
effects. Many materials, however, have a time-dependent be- atoms or movement of dislocations in the indentation stress
havior when placed under load and as a result, conventional field, the extent of which depends very much on the tem-
nanoindentation test methods may not provide an adequate perature. If we ignore effects arising from the formation of
estimation of material properties of interest. There are two cracks, permanent deformation in a material under indenta-
basic approaches to managing time-dependent behavior in tion loading is thus seen as arising from a combination of
nanoindentation testing. The first is the application of an os- instantaneous plasticity (which is not time-dependent) and
cillatory displacement or force, in which the transfer func- creep (which is time-dependent). A material which under-
tion between the load and displacement provides a method goes elastic and non-time-dependent plastic deformation
of calculating the storage and loss modulus of the material. is called elasto-plastic. A material that deforms elastically
In the second, the application of a step load or displacement but exhibits time-dependent behavior is called visco-elastic.
and subsequent measurement of depth (creep) or force as a A material in which time-dependent plastic deformation
function of time (relaxation) is used to calculate visco-elastic occurs is visco-plastic [1]. The term creep is often used to
describe a delayed response to an applied stress or strain
that may be a result of visco-elastic or visco-plastic de-
∗ Tel.: +61 2 9413 7544; fax: +61 2 9413 7457. formation. In a nanoindentation test, the depth recorded at
E-mail address: tony.cripps@csiro.au (A.C. Fischer-Cripps). each load increment will be, in general, the addition of that

0921-5093/$ – see front matter Crown Copyright © 2004 Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.msea.2004.04.070

A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82 75

Fig. 2. Displacement response for a step increase in load for (a) Voigt
model, (b) Maxwell model. f(h) = h2 for cone, f(h) = h3/2 for sphere.

The square-bracketed term in Eq. (1) represents the
displacement-time response of the mechanical model to a
Fig. 1. (a) Three-element Voigt spring and dashpot representation of step increase in load. A step increase in applied load re-
a visco-elastic material (delayed elasticy), (b) Maxwell representation sults in an initial elastic displacement (t = 0) followed by a
of a visco-elastic material (steady creep), (c) four-element combined delayed increase in displacement to a maximum value at t
Maxwell–Voigt model. = ∞ as shown in Fig. 2(a). It should be noted that Eq. (1)
applies to the case of a rigid indenter in which case the
symbol E∗ is the combination of the elastic modulus and
due to the elastic-plastic properties of the material and that Poisson’s ratio of the specimen material (E∗ = E/(1 − ν2 ))
occurring to due creep, either visco-elastic or visco-plastic. and not the combined modulus of the indenter and speci-
Time dependent properties of materials are conventionally men as is normally the case. The numerical factor 3/4 in
analyzed in terms of mechanical models such as those shown Eq. (1) differs from that usually seen in reference literature
in Fig. 1. The elastic response of such a model is quanti- due to our working with the elastic modulus E∗ rather than
fied by what we call the storage modulus. The fluid-like re- the shear modulus G. The justification for the change in
sponse is quantified by the loss modulus. In rheology, the variable being that in indentation loading, the greater pro-
material behavior is towards the fluid end of the spectrum portion of the deformation is hydrostatic compression and
where any elastic response, or the storage modulus, is dom- the materials we consider are more likely to be dominated
inated by the shear modulus of the material. In solids, such by solid-like properties than fluid-like behavior.
as those usually tested in nanoindentation, the material be- A similar approach is appropriate for the case of a conical
havior tends towards a predominantly elastic response. In indenter in which we obtain, for the case of the three-element
an indentation test, the nature of the loading is a complex Voigt model
mixture of hydrostatic compression, tension, and shear. Un-
1 1 −tE∗ /η
like a fluid, the storage modulus in this case contains contri- h2 (t) = Po cot α ∗ + E∗ (1 − e
2 ) (2)
butions from all three of these types of materials response. 2 E1 2
In the present work, we shall, in the interests of simplicity, For the case of a Maxwell model Fig. 1(b), the
assume that the storage modulus is a measure of the con- time-dependent depth of penetration for a spherical indenter
ventional (tensile/compressive) elastic modulus in recogni- is given by
tion of the large component of hydrostatic stress in the in-
dentation stress field. The fluid-like response we shall call 3 Po 1 1
h32 (t) = √ ∗ + ηt (3)
“viscosity” although in practice, viscosity is usually fre- 4 R E1
quency and temperature dependent and not single-valued.
and for a conical indenter, we obtain
Radok [2], and Lee and Radok [3] have analyzed the
visco-elastic contact problem using a correspondence prin- 1 1
h2 (t) = Po cot α ∗ + ηt (4)
ciple in which elastic constants in the elastic equations of 2 E1
contact are replaced with time dependent operators [4]. For
From the above information, it is relatively easy to con-
the case of a rigid spherical indenter in contact with a ma-
struct equations for more complicated models by adding
terial represented by a three-element Voigt model as shown
components in series or parallel as would be done in
in Fig. 1(a), we can write that for a steady applied load
mechanical–electrical analogs conventionally applied to this
Po , the depth of penetration increases with time is given
type of modeling. For example, in the present work, we shall
by the well-known Hertz equation with the addition of a
also consider a four-element Maxwell–Voigt combination
time-dependent exponential
as shown in Fig. 1(c) which gives
3 Po 1 1 −tE∗ /η 3 Po 1 1 −tE∗ /η2 1
h32 (t) = √ ∗ + E∗ (1 − e
2 ) (1) h32 (t) = √ ∗ + E∗ (1 − e
2 )+ t (5)
4 R E1 2 4 R E1 2 η1

76 A.C. Fischer-Cripps / Materials Science and Engineering A 385 (2004) 74–82

1 1 −tE∗ /η2 1 ified in draft standard ISO14577. A simplified treatment of
h2 (t) = Po cot α ∗ + E∗ (1 − e
2 )+ t (6) nanoindentation creep is presented and a readily accessible
2 E1 2 η1
method of analysing Eqs. (1)–(4) in a manner suitable for
The equations given above are expressed in terms of the automation within a computer program is provided.
modulus E∗ which is the combination of elastic modulus
and Poisson’s ratio of the specimen. In the present work,
the elastic material properties of the specimen material are 2. Experimental
given in terms of E∗ as is done in conventional nanoinden-
tation analysis and η is the viscosity term that quantifies Several materials were selected for study. The first, a 1 m
the time-dependent property of the material. It should be re- thick film of high purity aluminum, a metal known to ex-
membered that Eqs. (1)–(6) assume a step increase in load hibit significant indentation creep. The second, a sample of
to Po and are expressed here in a form to be easily fitted to fused silica, a material in which creep is not expected to be
experimentally obtained creep data to provide values for E∗ significant, and the third, an 100–150 m thick polyurethane
and η. Should modeling of an arbitrary time displacement acrylic copolymer film, a material expected to exhibit sig-
response be required, a suitable superposition [5,6] can be nificant visco-elasticity the amount of which is dependent
employed but this is beyond the scope of the present work. upon the additive molecules. Two acrylic co-polymer ma-
There are several detailed theoretical studies of visco-elastic terials were tested, a baseline material and another with a
indentation creep available in the literature [7–12]. A popu- cross-linker added. Tests were done on a UMIS nanoinden-
lar motivation for such modeling is the behavior of materials tation instrument [26]. This instrument is characterized by a
at elevated temperatures [13,14]. Modeling of indentation real-time electronic force-feedback control loop that ensures
creep for nanoindentation applications has also widely re- the indenter load is held constant regardless of the depth of
ported in the literature. Such treatments focus on either penetration during the creep period.
constant load (creep [15]) or constant displacement (relax- Conventional nanoindentation load-displacement curves
ation [16]) or both [17–21]. Traditionally, intrinsic material were done along with step loading and hold periods at max-
properties are modeled in terms of spring and dashpot el- imum load. Analysis of the unloading data in the usual man-
ements under indentation loading. For example, Feng and ner yields values for modulus and hardness on the assump-
Ngan [22,23] applied a Maxwell two-element model to tion of no time-dependent response. Fitting to Eqs. (1)–(6)
the creep displacement at maximum load in a conventional for the types of models considered here was done using a
load-displacement response and determined an equivalent least squares method. For the Maxwell model, this is rel-
expression for the contact stiffness that included the creep atively straight-forward, there being only two unknowns.
rate expressed as a displacement over time. Their work For the three-element Voigt and four-element combination
illustrates and quantifies the forward going “nose” that model, a non-linear least squares method was used the de-
appears in the unloading curve in indentation experiments tails of which are given in the Appendix A. In this method,
in which creep is significant. Cheng et al. [17] applied a starting values of E∗ and η are required. The method allows
method of functional equations to the visco-elastic contact for the specification of a tolerance level for convergence and
problem in conjunction with a step increase in load, or also the specification of relaxation factors to be applied for
displacement, to a three-element Voigt model to provide each variable to prevent instability in the computations. The
equations for steady creep, or relaxation, for the case of a procedure for materials that exhibit significant creep is rela-
rigid spherical indenter in contact with both incompress- tively straight forward, convergence is rather rapid. For solid
ible and compressible materials. In contrast, in a recent like materials, such as the aluminum and fused silica tested
work, Oyen and Cook [24] presented a phenomenological here, it is necessary to reduce the value of the relaxation
approach which sought to include elasticity, viscosity, and factors progressively and to undertake many iterations.
plasticity in terms of modeling elements that represented In all the experiments reported here, testing was done in
the quadratic character of the contact equations rather than under very closely controlled laboratory conditions in which
the intrinsic properties of the specimen material. Those the ambient temperature was held at 21 ± 0.1 ◦ C. Each test
workers found a very good agreement between the predicted was performed after a long thermal soak period, and thermal
and observed load-displacement curves for the bounding drift was assessed before each experiment by monitoring
conditions of an elastic-plastic response (e.g. metals and the displacement output of the indentation instrument at the
ceramics) to visco-elastic deformation (e.g. elastomers). initial contact force. Thermal drift during all the experiments
The previous works mentioned here are characterized by was deemed to be negligible and no correction of the final
very formal constitutive equations and complex methods of data was made during the analysis. The creep times selected
solution. By way of contrast, in draft standard ISO14577 for this study are comparatively short compared to larger
[25], creep is simply expressed as a change in depth (or scale indentation testing. This ensures that any thermal drift
load) over time for fixed load, or fixed displacement loading. errors that might occur are kept to a minimum.
The intention of the present work is to fill the gap between For the aluminum specimen, a nominal 20 m radius
those more formal treatments discussed above and that spec- diamond spherical indenter was used to perform standard


Table 1 yielded E1 = 58.6 GPa, E2 = 1197 GPa and η = 1886 GPa s.
Values of elastic modulus, E, hardness, H, and total penetration depth, A least squares fit to the two-element Maxwell model, gave
ht , from the unloading response of conventional load-displacement curves
on a 1 m Al film deposited on silicon
E1 = 75.6 GPa and η = 28797 GPa s. The value of radius
used for the fitting was adjusted according to the area func-
Hold (s) E (GPa) H (GPa) ht (nm)
0 109 0.932 73.8 tion of the indenter and at the depth of penetration measured,
10 97.8 0.928 74.7 translated into an actual radius of 36 m. The displacement
20 112 0.926 76.7 response function using the values in Eq. (1) is shown as a
40 105 0.926 77.4 full line in Fig. 3.
80 102 0.922 79.9
The data shown in Fig. 3 was taken with some consid-
eration for minimizing the dynamic response of the mea-
surement instrument. The displacement data was taken af-
load–unload indentation tests with varying hold periods at ter amplification from the depth sensor but before the nor-
maximum load to determine the effect of creep on the com- mal filtering circuitry so that any time-related delays with
puted values of modulus and hardness. The actual radius of the electronic filtering would not affect the results obtained.
the indenter at the penetration depth for each experiment The data therefore contains a level of electronic noise that
was determined by calibration against a fused silica speci- would ordinarily not be measured in a nanoindentation test.
men and used in the modeling procedure. It was found that The application of the 10 mN load took approximately 1 s,
for the indenter used here, there was a substantial flatten- the fastest rate available with the test instrument.
ing of the radius at small penetration depths and a sharpen- For the fused silica specimen, a corner cube indenter was
ing of the radius at higher depths. This is not unusual with used in both a conventional load–unload indentation test and
sphero-conical indenters used in nanoindentation work. The also in step loading and hold. Conventional analysis of the
results for modulus and hardness with varying creep times unloading yielded E = 74.2 GPa and H = 9.5 GPa at a depth
are shown in Table 1. A Poisson’s ratio of ν = 0.35 was of penetration of 464 nm for a maximum load of 10 mN. A
assumed for aluminum to extract the specimen modulus E value of ν = 0.17 was used to extract the specimen modulus
from the combined modulus E∗ . from the combined modulus. These results are in reason-
Further, tests with a step load followed by a hold period able agreement with the commonly accepted values of 72.5
were performed and the resulting data analyzed using the and 9.2 GPa, respectively. The effective cone angle at this
Eqs. (1) and (3). A step load of 10 mN was applied to the depth was found to be 57◦ from the calibrated area function
same indenter and held constant for 20 s. The change in depth of the indenter. For a step loading to 10 mN and hold pe-
as a function of time is shown in Fig. 3. The depth changed riod for 20 s, the resulting change in displacement as a func-
from an initial value of 70.4–78.6 nm over the 20 s hold pe- tion of time was analyzed using Eq. (2) using the non-linear
riod. A non-linear least squares analysis of the hold period least squares method. A significant number of iterations
data for a three-element Voigt model according to Eq. (1) and adjustments to the relaxation factors was required to
obtain convergence. The results of the fitting yielded E1
= 36.33 GPa, E2 = 1.27 × 108 GPa and η = 8150 GPa s.
Linear least squares fitting for the two-element Maxwell
model yielded E1 = 36 GPa and η = 52816 GPa s. A com-
parison of the experimental and fitted data for the hold pe-
riod is shown in Fig. 4. The relatively large amount of scatter
(≈ ±1 nm) in Fig. 4 arises from taking the depth readings
before any filtering to ensure the most representative mate-
rial time-related response. As can be seen from the results
above, there is a significant difference in the value for E1
provided by the creep analysis compared to that obtained
with the conventional unload analysis. This arises from the
finite loading time of the indenter to the maximum load and
more discussion on this is given below.
The corner cube indenter was selected for this material
since experience with a spherical indenter showed that it
was difficult to obtain any creep response in this material.
A cube-corner indenter was thought to provide a very high
Fig. 3. Creep response for hold period of 20 s at 10 mN on a 1 m Al film level of indentation stress so as to induce a visco-elastic
on silicon with a spherical indenter 20 m nominal radius (36 m actual
radius). Data points show experimental results. The solid line shows the
or visco-plastic response more readily than would be pos-
fitted response according to Eq. (1) (three-element Voigt model). The sible with a blunter type of indenter. With the cube cor-
dotted line shows the response according to Eq. (3) (two-element Maxwell ner indenter, reloading the impression in this material after
model). some minutes from the initial testing showed a completely


Table 2
Results from conventional nanoindentation analysis on the unloading
portion of load–unload tests and creep tests at 1 mN for 10 s for indentation
with a nominal 20 m radius spherical indenter on (a) acrylic co-polymer
film and (b) acrylic co-polymer film with cross-linker additive (both films
were approximately 100–150 m)
Polymer (a) Polymer (b)

Load/unload test to 1 mN at constant ht = 1680 nm ht = 313 nm
strain rate 20%/s
E (GPa) 0.154 1.209
H (GPa) 0.0086 0.0571
ht (nm) 1680 313
Creep test at step load to 1 mN, ht = 2000 nm ht = 373 nm
hold for 10 s
Two-element Maxwell model
E1 (GPa) 0.0896 0.9414
η (GPa s) 1.687 41.25
Fig. 4. Creep response for hold period of 20 s at 10 mN on a fused
silica with a corner cube indenter with an effective cone angle of 57◦ Three-element Voigt model
as determined from the indenter area function. The solid line shows the E1 (GPa) 0.163 1.033
fitting according to Eq. (2) (three-element Voigt model) while the dotted E2 (GPa) 0.103 2.732
line shows the fitting according to Eq. (4) (two-element Maxwell model). η (GPa s) 0.135 11.18
Four-element Maxwell–Voigt model
E1 (GPa) 0.175 1.055
flat creep response during a hold period indicating that the E2 (GPa) 0.136 4.703
creep observed here is related to visco-plasticity rather than η1 (GPa s) 0.696 7.688
visco-elasticity. η2 (GPa s) 2.856 72.6
Load-displacement response and a creep test at step
load of 1 mN for 10 s were performed on the two acrylic modulus E from the combined modulus E∗ . For these tests,
co-polymer materials to a maximum load of 1 mN with a the load was applied over a period of X and Y s.
20 m radius spherical indenter. At the depths of penetra-
tion in this material, the actual radius of the indenter tip was
estimated to be 13.5 m from the calibrated area function of 3. Discussion
the indenter. The load-displacement curves were performed
at a constant strain rate of 20%/s and are shown in Fig. 5. As mentioned in Section 1, the depth recorded at each
The results are given in Table 2 along with the moduli and load increment in a nanoindentation test will be that arising
viscosity estimations from least squares fitting to the creep from elastic-plastic, visco-elastic and visco-plastic deforma-
response from a step loading to 1 mN and hold over 10 s. tions. In conventional nanoindentation testing, the instanta-
The creep responses for the two materials are shown in neous elastic and plastic deformations are usually consid-
Fig. 6 along with the fitted curves from Eqs. (1), (3) and ered. Elastic equations of contact are applied to the unload-
(5). A value of ν = 0.4 was used to extract the specimen ing data to find the depth of the circle of contact at full load

Fig. 5. Load–unload response for indentation with a nominal 20 m radius spherical indenter on (a) acrylic co-polymer film and (b) acrylic co-polymer
film with cross-linker additive.


Fig. 6. Creep response for hold period of 10 s at 1 mN on (a) acrylic co-polymer film and (b) acrylic co-polymer film with cross-linker additive with a
13.5 m spherical indenter as determined from the indenter area function. Experimental data is shown as data points. Line plots indicate fitting to two-,
three- and four-element models.

(under elastic-plastic conditions). If there is a visco-elastic visco-plastic behavior. In the case of aluminum, the effect is
or visco-plastic response (i.e. creep), then this analysis is in- aggravated due to piling-up of material around the edge of
valid. In extreme cases, the unloading curve has a negative the contact area. In instrumented indentation tests, the piled
slope which, if the standard unloading analysis is applied, up material serves to make the specimen appear harder and
results in a negative elastic modulus [22,27]. stiffer because conventional analysis techniques do not ac-
When fitting creep curves to mechanical models, we must count for it.
be sure that we distinguish between visco-elasticity and In the present work, the loading time for the step load
visco-plasticity. The models shown here are appropriate for was limited by the time-response of the electronics of the
visco-elastic deformation but may be shown to provide some test instrument. An attempt was made to gauge the effect
information for visco-plastic deformation. If a step load is of the loading time for the case of the cube corner indenter
applied to an indenter in contact with a material and the on fused silica by performing a number of creep indenta-
resulting depth of penetration monitored, then a response tion tests while attempting to increase the loading time by
similar to that shown in Fig. 2(a) or (b) may be obtained. altering the time response of the instrument circuitry. Faster
Fitting the appropriate equation (Eqs. (1)–(4)) to this data loading times could be achieved, but it was found that the
yields values for the moduli and viscosity terms. The ac- indenter penetrated the sample more deeply due to over-
curacy of the results so obtained depends upon the rapid- shoot in the force servo-control feedback loop. Attempts to
ity with which the step increase in load is applied and the reduce the overshoot resulted in a more damped response
time-dependent nature of the specimen material. In practice, which increased the time taken to achieve the maximum load
an increase in load is applied over a finite time period within which also, as discussed above, increased the penetration
which, for elastic-plastic materials, plastic deformation can depth due to plastic deformation in the specimen. These at-
occur quite rapidly and this causes the initial step response tempts showed that for highly elastic materials, a creep anal-
in displacement to be greater than that predicted, particu- ysis employing a step increase in load is likely to provide
larly when a sharp indenter is used. The resulting value of values of E1 only to within an order of magnitude. In this
modulus can be very much less than the nominal modulus case, conventional nanoindentation would be more appro-
of the material. For a step load with a spherical indenter priate for the measurement of elastic properties. However,
(to reduce the possibility of time-independent plasticity) on measurements of the time-dependent properties of such a
a material with a significant viscous response, the resulting material are not so affected by the non-instantaneous step
analysis is likely to result in a reasonable measurement of application of load and the measurement of the viscosity
the visco-elastic properties of the material. The significance terms would be expected to be reliable. In the extreme case
of this can be observed by comparing the values of mod- of solid-like materials (e.g. the fused silica specimen tested
ulus obtained from conventional unloading curve analysis here), the procedure may be useful in determining a quantita-
to that obtained from the creep curve fitting. For aluminum tive account of the time-dependent nature of the deformation
and fused silica, the moduli E1 from the creep curve fitting under extremely high contact pressures which would ordi-
procedures are very much less than expected because of the narily be inaccessible in conventional tensile testing where
plastic deformation arising during the step loading. This is fracture of the specimen occurs before plastic deformation.
because the models assume an initial instantaneous elastic For visco-elastic property measurements, the creep analysis
response (i.e. the models represent a visco-elastic deforma- procedure presented here is more suitable for use on ma-
tion) whereas in these materials, there is an elastic-plastic terials in which time-dependent behavior represents a sig-
response, and in the case of aluminum, a relatively strong nificant contribution to the overall deformation (such as the


polymer materials tested here). In the case of the polymer the deformation is desired. The scope of the present pa-
specimens, a relatively low load of 1 mN was found to pro- per is to present a simple phenomenological approach only.
vide a reasonably large penetration depth in each case and A comparison of the fitted creep response curves and the
requirement for a more ideal step-like application thus more experimental data presented in Figs. 3, 4 and 6 demon-
easily accomplished. strate that the mechanical models used may not precisely
For the case of the two polymeric materials tested here, the match the response of the specimens, particularly in the
specimen with the added cross-linker shows a much stiffer case of the two-element Maxwell model. The three-element
elastic and more viscous response compared to the base ma- Voigt model provides a reasonable fit while the four-element
terial. This is expected as the cross linker serves to restrict model (Fig. 6) shows a very good fit. The physical signif-
the motion of molecular chains under the applied shear stress icance of the elements within a model depends upon the
in the indentation stress field. In general, the response of a microstructural characteristics of the specimen material.
particular specimen material may be affected by indentation In the present case, no weighting scheme was used other
size effects arising from strain-gradient plasticity, the pres- than an adjustment to the values of the relaxation factors.
ence of oxide layers, surface roughness, etc. Such effects are However, the non-linear least square fitting procedure is gen-
beyond the scope of the present work but should be consid- eral enough to allow the addition of more elements as de-
ered as possible sources of errors or variations in the data, sired. The fitted curves shown here represent a minimization
especially in the case of crystalline solids. of the sum of the squares of the differences between the fit-
The load–unload curves shown in Fig. 5 deserve some ted and actual data, but more control is possible through the
comment. The polymeric materials tested here were selected use of the terms wi given in the Appendix A. Using these
on the expectation of a significant visco-elastic response (in factors, the differences between the fitted and actual data
contrast to an elastic-plastic response). The curves shown in can be weighted in favor of data at one or other ends of the
Fig. 5 support this, especially for the cross-linked material. range of values as desired.
Although there is a readily identifiable plastic deformation Some workers [27,31] have studied the effect of specimen
as evidenced by the area enclosed between the loading and creep on the values of modulus and hardness obtained using
unloading curves, the elastic recovery of the material is quite conventional methods of analysis of the unloading response.
substantial. This implies that the energy dissipation within The general conclusion is that for materials which exhibit
these types of materials is substantially a result of viscous creep during an indentation test, the modulus so calculated
losses (in the sense of visco-elasticity) and not plastic defor- from the unloading response is not reliable if the hold period
mation (in the sense of elasto-plasticity). Similar behavior at maximum load is too short due to bowing or “nose” of
in various materials has been previously reported in the lit- the unloading response to larger depth values resulting from
erature [28,29] but has not in general been satisfactorily ex- creep. Briscoe et al. [31] introduced a 10 s hold period into
plained. If we recognize the limitations of the method, then their tests on polymeric specimens to eliminate the nose in
the values for E2 and η, in the case of the three-element Voigt the unloading data. Chudoba and Richter [27] found that the
model, may still have validity for visco-plastic contact since, hold period at maximum load has to be long enough such
as shown in Fig. 2, they influence only the time-dependent that the creep rate has decayed to a value where the depth
character of the deformation and not the initial response to increase in 1 min is less than 1% of the indentation depth.
the step loading. A decision as to the nature of the speci- According to Chudoba and Richter, allowing creep to pro-
men material with regard to its elasto-plastic, visco-plastic or ceed to relative completion and then obtaining the unload-
visco-elastic character can thus in some cases be made upon ing data would provide a value of unloading stiffness dP/dh
the observed shape of the conventional load-displacement that would occur at the increased depth free from the effect
response. of creep. Feng and Ngan [22] draw a similar conclusion and
While the use of simple mechanical models such as those show how a value of dP/dh can be obtained using shorter
shown in Fig. 1 allow some comparison to be made between hold periods if the unloading slope is corrected by a factor
specimens, it should be noted that they offer very little in which is dependent upon the creep rate and the unloading
terms of a basic understanding of the physical mechanisms load rate.
involved in the deformation. The mathematical models in
Eqs. (1)–(6) are simply standard elastic equations with a
time-dependent terms added to represent the fluid-like be- 4. Conclusion
havior of the material. They are models only with no real
physical significance. They serve only to give some quan- The intention of the present work is to provide a simple
titative description of mechanical events. Time-dependent and accessible method of obtaining a quantitative measure
behavior often depends on the strain-hardening characteris- of the elastic and viscous properties of materials from inden-
tics of the material which in turn, depend upon microstruc- tation creep curves. The present work is not intended to of-
tural variables. Various constitutive laws [30] have been pro- fer a rigorous account of indentation creep or materials con-
posed that apply to many different types of materials and stitutive behavior. Three representative classes of materials
these should be investigated if a more detailed account of were tested: a highly elastic ceramic, a soft metal and two


visco-elastic polymer systems. For highly elastic solid-like yi − Zi at each data point i can be weighted by a factor wi to
materials, the results using the creep method for the elas- reflect the error associated in the observed values yi . Thus,
tic modulus are very much less than that expected due to the sum of the squares is expressed as
the sensitivity of the technique to the non-deal step-like ini-
N
tial application of load. The technique may however provide
X2 = wi [yi − Zi ]2
quantitative information about the plastic behavior of these
i=1
materials which would not ordinarily be accessible through   2
conventional tensile or compressive tests where fracture usu- N r
δZi
o
ally occurs before deformation. For visco-elastic materials, = wi yi − Zi +
o
δaj 
δaj
the technique is expected to be useful for the measurement i=1 j=1
of both viscous and elastic properties. Lower loads may be  2
N r
used to obtain reasonable penetration depths thus allowing a δZi
o
= wi (yi − Zi ) −
o
δaj  ,
step-like application of load to be more easily accomplished. δaj
i=1 j=1
Depending on the material tested, the results can be consid-  2
ered a figure of merit, or absolute values of elasticity and N r
δZi
o
viscosity. The theoretical work presented here is based on yi = yi − Zi ,
o
X2 = w i  yi − δaj 
δaj
standard phenomenological models and the fitting procedure i=1 j=1
can be readily automated. (A.5)
The weighting factor wi for the present application can be
Acknowledgements simply the magnitude of yi on the assumption that the error
at each data point is inversely proportional to the magnitude
The author thanks Avi Bendavid for supplying the high of the data at that point.
purity aluminum film, Jeffrey T. Carter for supplying the The objective is to minimise this sum with respect to the
polymer films used in this study, and A.H.W. Ngan and an values of the error terms δaj , thus we set the derivative of
anonymous referee for useful comments. X2 with respect to δaj to zero

δX2
Appendix A. Linear approximation, non-linear least = 0,
squares δ(δaj )
  
N r
δZi
o δZi
o
Let Zi be a function that provides fitted values of a de- 0= wi  yi − δaj   (A.6)
δaj δaj
pendent variable yi at each value of an independent variable i=1 j=1
xi . Zi can be a function of many parameters a0 , a1 , . . . , ar .
This expression can be expanded by considering a few
Zi = Zi (xi : a0 , a1 , a2 , . . . , aj , . . . , ar ) (A.1) examples of j. Letting j = 1, we obtain
It is presumed that initial values or estimates of these N
δZi
o
parameters are known and that the desired outcome is an wi y i
δa1
optimisation of the values of these parameters using the i=1
method of least squares. The true value of the parameter aj N 2 N
δZi
o δZi δZi
o o
is found by adding an error term δaj to the initial value aj .
o = δa1 wi + δa2 wi
δa1 δa1 δa2
i=1 i=1
aj = aj + δaj
o
(A.2) N
δZi
o δZi
o
Thus, the function Zi becomes + · · · + δar wi (A.7)
δa1 δar
i=1
Zi = Zi (xi : a1 , a2 , a3 , . . . , ar )
o o o o o
(A.3)
At j equal to some arbitrary value of k, we obtain:
If the errors δaj are small, then the function Zi can be
N
expressed as a Taylor series expansion δZi
o
wi y i
r
δZi
o δak
i=1
Zi = Zi
o
+ δaj (A.4)
δaj N
δZi δZi
o o N
δZi δZi
o o
j=1
= δa1 wi + δa2 wi + ···
δak δa1 δak δa2
This is a linear equation in δaj and is thus amenable to i=1 i=1
multiple linear least squares analysis. N 2 N
δZi
o δZi δZi
o o
Now, by least squares theory, we wish to minimise the sum + δak wi + · · · + δar wi
δak δak δar
of the squares of the differences (or residuals) between the i=1 i=1
observed values yi and the fitted values Zi . The differences (A.8)


In matrix notation, the sums for each error term δa from [3] E.H. Lee, J.R.M. Radok, Trans. ASME Series E, J. Appl. Mech. 27
1–r is expressed (1960) 438–444.
[4] K.L. Johnson, Contact Mechanics, Cambridge University Press, 1985.
[5] L. Cheng, X. Xia, W. Yu, L.E. Scriven, W.W. Gerberich, J. Polym.
[Yj ] = [Ajk ][Xj ], Sci. B 38 (2000) 10–22.
    
Y1 A11 . . . A1r X1
[6] M. Sakai, Phil. Mag. A 82 (10) (2002) 1841–1849.
[7] P.M. Sargent, M.F. Ashby, Mater. Sci. Technol. 8 (1992) 594–601.
Y    
 2  . . . . .   X2  [8] R. Hill, B. Storåkers, A.B. Zdunek, Proc. R. Soc. Lond. A423 (1989)
    
 Yk  =  . . Ajk . .   Xk  (A.9)
301–330.
     [9] R. Hill, Proc. R. Soc. Lond. A436 (1992) 617–630.
    
 .   . . . . .  .  [10] B. Storåkers, P.-L. Larsson, J. Mech. Phys. Solids 42 (2) (1994)
307–332.
Yr Ar1 . . . Arr Xr [11] S.Dj. Mesarovic, N.A. Fleck, Proc. R. Soc. Lond. A455 (1999)
2707–2728.
where [12] Y.-T. Cheng, C.-M. Cheng, Phil. Mag. Lett. 81 (2001) 9–16.
N [13] T.R.G. Kutty, C. Ganguly, D.H. Sastry, Scripta Mater. 34 (12) (1996)
δZi
o
1833–1838.
Yj = wi yi ,
δaj [14] M. Sakai, S. Shimizu, J. Non-Cryst. Solids 282 (2001) 236–
i=1 247.
N [15] W.B. Li, R. Warren, Acta Metall. Mater. 41 (10) (1993) 3065–
δZi δZi
o o
Ajk = Akj = wi , Xj = δaj (A.10) 3069.
δaj δak [16] S. Shimizu, Y. Yanagimoto, M. Sakai, J. Mater. Res. 14 (10) (1999)
i=1
4075–4085.
and [17] L. Cheng, L.E. Scriven, W.W. Gerberich, Mater. Res. Symp. Proc.
522 (1998) 193–198.
yi = yi − Zi
o
(A.11) [18] A. Strojny, W.W. Gerberich, Mater. Res. Soc. Symp. Proc. 522 (1998)
159–164.
The solution is the matrix X that contains the error terms [19] S.A. Syed Asif, J.B. Pethica, J. Adhes. 67 (1998) 153–165.
to be minimised. Thus [20] X. Xia, A. Stronjy, L.E. Scriven, W.W. Gerberich, A. Tsou, C.C.
Anderson, Mater. Res. Symp. Proc. 522 (1988) 199–204.
[Xj ] = [Ajk ]−1 [Yj ] (A.12) [21] K.B. Yoder, S. Ahuja, K.T. Dihn, D.A. Crowson, S.G. Corcoran, L.
Cheng, W.W. Gerberich, Mater. Res. Symp. 522 (1998) 205–210.
When values of δaj are calculated, they are added to the [22] G. Feng, A.H.W. Ngan, J. Mater. Res. 17 (3) (2002) 660–668.
initial values aj to give the fitted values aj
o [23] A.H.W. Ngan, B. Tang, J. Mater. Res. 17 (10) (2002) 2604–
2610.
aj = aj + Lδaj
1 o
(A.13) [24] M.L. Oyen, R.F. Cook, J. Mater. Res. 18 (1) (2003) 139–150.
[25] ISO14577, Metallic materials—instrumented indentation test for
The process may then be repeated until the error terms δaj hardness and materials parameters, ISO Central Secretariat, 1 rue de
become sufficiently small indicating that the parameters aj Varembé, 1211 Geneva 20, Switzerland.
[26] CSIRO Division of Telecommunications and Industrial Physics, P.O.
have converged to their optimum value. L in Eq. (A.13) is a Box 218, Lindfield, NSW 2007, Australia.
relaxation factor that is applied to error terms to prevent in- [27] T. Chudoba, F. Richter, Surf. Coat. Tech. 148 (2001) 191–198.
stability during the initial phases of the refinement process. [28] M.V. Swain, J.S. Field, Phil. Mag. A 74 (5) (1996) 1085–
1096.
[29] M.W. Seto, K. Robbie, D. Vick, M.J. Brett, L. Khun, J. Vac. Sci.
Technol. B 17 (5) (1999) 2172–2177.
References [30] D. François, A. Pineau, A. Zaoui, Mechanical Behaviour of Materials,
Kluwer Academic Publishers, The Netherlands, 1998.
[1] A.C. Fischer-Cripps, Nanoindentation, Springer-Verlag, NY, 2002. [31] B.J. Briscoe, L. Fiori, E. Pelillo, J. Phys. D 31 (1998) 2395–
[2] J.R.M. Radok, Q. Appl. Math. 15 (1957) 198–202. 2405.

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