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Higher accuracy analysis of instrumented indentation data obtained with pointed indenters
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2008 J. Phys. D: Appl. Phys. 41 215407
(http://iopscience.iop.org/0022-3727/41/21/215407)
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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 41 (2008) 215407 (16pp) doi:10.1088/0022-3727/41/21/215407
Higher accuracy analysis of instrumented
indentation data obtained with pointed
indenters
T Chudoba
1
andNMJennett
2
1
ASMEC Advanced Surface Mechanics GmbH, Bautzner Landstr. 45, 01454 Radeberg, Germany
2
National Physical Laboratory, Hampton Road, Teddington, Middlesex, TW11 0LW, UK
Received 11 June 2008, in final form 1 September 2008
Published 14 October 2008
Online at
stacks.iop.org/JPhysD/41/215407
Abstract
We review current practice for describing force–displacement curves from pointed indenters,
highlighting the consequences of the simplifications normally adopted. We derive two
corrections, the ‘variable epsilon factor’ and the ‘radial displacement correction.’ These are
especially important for highly elastic materials such as fused silica where the combined
corrections can amount to 13% in the contact area, significantly increasing the accuracy of
hardness and modulus results. In contrast, the so-called beta factor has minor importance. We
compare our analytical results with finite element (FE) calculations and experimental results.
Indenter area functions, obtained using the corrections, agree well with independent direct
measurements by a traceably calibrated metrological atomic force microscope (AFM). Further
formulae are derived to calculate the complete force–displacement curve of conical indenters
and the indentation elastic and total energy. These formulae immediately identify a physical
material limit above which a cone cannot generate plastic deformation; for a Berkovich
indenter this is a hardness-to-modulus ratio of 0.18.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
An important driver for the development of instrumented
indentation testing was the fact that smaller indentations were
required to measure the properties of engineered surfaces
and coatings. These indents were becoming too small for
optical determination of the indentation area to be achieved
with the necessary accuracy. It was realized, however, that,
instead of measuring the cross sectional area of the indentation,
indentation depth could be measured automatically with
high accuracy and resolution in the nanometre or even sub-
nanometre range. This could then be converted to an area
if the shape of the indenter was known. Furthermore, an
instrumented depth measurement is operator independent.
Consequently, instrumented indentation force–displacement
data have been used for about 30 years to determine the
mechanical properties of surfaces and thin films; the first
publications known by the authors were published in 1973 by
Ternovskij, Alekhin and Bulychev et al in the former Soviet
Union [1, 2] and by Fr
¨
ohlich and Grau in the GDR [3, 4].
However, it was the pioneering work of Loubet in 1984 [5],
Doerner and Nix in 1986 [6] and of Oliver and Pharr in 1992
[7] that enabled interpretation of force–displacement curves
in a way that separated the elastic and plastic contributions
of the deformation. This enabled a correction for the non-
ideal shape of the tip and derivation not only of hardness
but also of modulus values. Since then, many attempts have
been made to improve the accuracy of indentation analysis
and to introduce a number of corrections. An important
step in international consensus along the way to developing
instrumented indentation techniques as a standard analysis
tool and to achieving comparability of hardness numbers was
the international standard ISO 14577 [8] for the testing of
bulk materials, to which additional parts are continuing to be
added, most recently to provide for the indentation of coatings
(ISO 14577 part 4 published in 2007).
In this work, we review current practice for description
of the force–displacement curves from pointed indenters. In
the process, we demonstrate the derivation of the analytical
framework behind the current implementation of the Oliver
and Pharr [7] analysis and briefly describe some previously
proposed improvements. Our purpose is to identify and
0022-3727/08/215407+16$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 1. Geometry of an elasto-plastic deformation with a conical
indenter (a) under force, F , where α is the angle between indenter
surface and sample surface, A
c
is the projected area of contact,
radius a, which occurs at a distance h
c
from the indenter tip, h
max
is
the maximum indentation depth and h
s
is the bowing of the surface,
assumed to be elastic. (b) After force removal where h
0
is the
remnant indentation depth.
highlight the necessary simplifying assumptions and their
consequences. We then derive some improvements that may
increase the accuracy of hardness and modulus results. We
concentrate on pointed indenters, i.e. pyramids or cones,
because the self-similarity of the geometry simplifies matters.
Finally, our proposed extensions to the standard indentation
analysis are compared with finite element (FE) calculations
and validated by experimental data. We demonstrate that the
functions of indenter area of contact versus indentation depth
(i.e., the indenter ‘area function’), when obtained by indenting
different materials, only agree with each other and with the real
shape of the indenter (as measured directly by a metrological
atomic force microscope (AFM)) if additional corrections are
applied to the indentation analysis.
2. Derivation of the model
2.1. Basic assumptions of the standard indentation analysis
Consider a homogeneous material with depth independent,
constant hardness and Young’s modulus indented by a conical
indenter. For pyramidal indenters an effective contact radius
a
eff
=
A
c
π
. (1)
can be defined and the problem is reduced to an equivalent
conical one (only pyramids with equal side lengths are
considered).
Figure 1(a) shows an indentation with a conical indenter
at the time of maximum force and again after force removal,
figure 1(b). The maximum displacement measured, h
max
,isa
combination of the total deformation of the sample surface plus
the deformation of the indenter itself. The latter is considered
later but, for the moment, we shall assume a rigid indenter for
which only the displacement in the sample has to be considered.
The absolute displacement of the sample consists of the elastic
part h
e
and the plastic part h
p
with
h
max
= h
e
+ h
p
= h
c
+ h
s
(2)
and the hardness of the material can be described by the known
shape of an ideal conical indenter and a suitable definition of
indentation hardness H such as
H =
F
A
c
=
F
πa
2
, (3)
where F is force and A
c
is the projected contact area.
It therefore follows that, for a rigid indenter:
h
c
= a tan α = tan α
√
F
√
π ·
√
H
=
A
c
π
tan α, (4)
where α is the angle between the sample surface and the
indenter surface. For a cone with the same depth-to-area ratio
as a Vickers indenter, α is 19.7
◦
.
Similarly, if S is the gradient of the tangent to the
force-removal curve at maximum force and denotes the
contact stiffness of the indentation, ISO14577 gives the elastic
modulus of the material as
E =
√
π
2
·
S
√
A
c
. (5)
In both cases, the result depends upon an accurate input
of the projected area of contact. This requires an accurate
determination of the contact depth, h
c
, and the correct function
to relate that to the area of contact. (Note that it is assumed that
the contact area A(h
c
) is identical to the projected area of the
indenter at a distance h
c
from the tip when directly measured
under zero force, e.g. by a metrological AFM.) Since h
c
is
not directly measured, it is necessary to derive it by the use
of a suitable elastic contact model. The following simplifying
assumptions are used in current analyses.
(1) The projected contact area under applied force and the
projected area of the impression after force removal are
equal, so that, to a first approximation, there is only
vertical elastic recovery within the contact area and the
diameter of the indentation does not change when the force
is removed.
(2) Plastic and elastic deformation do not influence each
other, so that Young’s modulus of the plastically deformed
material is the same as that for undeformed material. The
elastic deformation of the surface is therefore the same
with or without plastic deformation (for an equal contact
area).
(3) Surface roughness, as well as pile-up or sink-in of the
material around the indenter, may be neglected.
None of these assumptions is absolutely true but they are
commonly used because they enable a great simplification of
the analysis of the contact problem. Assumption (1) allows
direct comparison of instrumented indentation hardness with
that of Vickers hardness. Assumptions (1) and (3) imply that
the plastic depth and contact depth are equal: h
c
= h
p
.If
2
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
we know the true shape of the tip (as we shall assume here),
then the contact area A
c
(and so, using assumption (1), the
remaining area of the impression) can be derived from the
indentation depth h
max
, provided we are able to determine
the elastic bowing, h
s
, of the free surface outside the contact
zone. The absolute elastic deformation of material directly
under the indenter is
h
e
= h
s
+ (h
c
− h
0
) = h
max
− h
0
, (6)
where h
0
= final depth after force removal and (h
c
− h
0
) is
the elastic recovery of the material directly under the indenter.
This elastic recovery acts to change the impression aspect
ratio during force removal, but does not need to be known
to calculate the contact depth, h
c
, since this can be obtained
from h
max
and h
s
.
2.2. Review of the determination of elastic surface
displacement for the case of purely elastic deformation
In addition to the deformation of the material directly in contact
with the indenter, the free surface on either side of the contact
also deforms elastically (as illustrated in figure 1). To obtain
the depth of elastic deformation of the surface we return to
considering wholly elastic deformations. We know from the
theory of elasticity that the force–displacement relationship,
F (h), for the centre of an indentation by a rigid cone of angle
α is the power function
F =
2
π
·
E
∗
tan α
· h
m
, where
E
∗
=
E
(1 − ν
2
)
and m = 2 (7a)
and for a sphere is
F =
4
3
· E
∗
·
√
R · h
m
where m = 3/2(7b)
where m is a geometry dependent exponent, E is Young’s
modulus, ν is Poisson’s ratio of the indented material, R is
the radius of the sphere and h is the indentation depth (depth
co-ordinate), which is purely elastic here. The first derivative
dF/dh at h
max
is the contact stiffness S and can be determined
from a tangent to the F –h curve at F
max
. Using the force–
displacement formulae (7a) and (7b), the ratio of applied force
to contact stiffness is therefore h/m (i.e. h/2 for a cone and
2h/3 for a sphere). Now, the vertical displacement of any
position on the surface by a cone is given by Sneddon [9, 10]as
w(r) =
2
π
π
2
−
r
a
· a · tan αr a,
w(r) =
2 · h
π · a
·
a · sin
−1
a
r
− r +
r
2
− a
2
r>a, (8)
where w(r) is continuous at r = a, r is the radial surface
position, h is the indentation depth and a is the contact radius.
Using r = a at the contact edge and r = 0 in the centre we
can obtain the ratio h
s
/h
max
= 1–2/π. Sneddon also gives
the solution for a paraboloid of revolution as the Hertzian
approximation for a sphere with h/R 1. The ratio for a
Table 1. Comparison of elastic deformation parameters for different
indenter geometries. h
s
/h
max
is obtained from equation (8), m is
defined by the indenter geometry and ε is calculated using
equation (10).
Indenter/pressure h
s
/h
max
mε
Flat punch 1 1 1
Sphere, paraboloid 0.5 3/2 0.75
of revolution
Cone 1–2/π = 0.3634 2 0.7268
Constant pressure, 2/π = 0.6366 1 0.6366
circular boundary
paraboloid of revolution (and for a sphere) is 1/2. The product
of this ratio and the exponent m is the so-called epsilon factor,
ε, introduced by Oliver and Pharr [7]. It is given in table 1
for some indenter geometries. Finally, we obtain the contact
depth
h
c
= h
max
− ε ·
F
S
, (9)
where in general
ε = h
s
1
F
dF
dh
=
h
s
h
h
F
dF
dh
so that ε = m ·
h
s
h
max
(10)
and m is again the exponent as in equations (7a), (7b) and
table 1. The contact area can be calculated from the known
geometry of the tip, i.e. the area function A(h
c
). (Again,
it should be noted that, if the assumptions in section 2.1
are correct, this area function should be identical to that of
an indenter under zero force when measured directly, e.g.
by a metrological AFM). The results for a constant pressure
applied within a circular boundary as given by Timoshenko and
Goodier [11] are also included as the bottom line in table 1 for
comparison. In this case, the exponent m is only unity if the
size of the area, where the constant pressure acts, does not
change during increasing or decreasing applied force.
In a real measurement, the indenter itself is also elastically
deformed. This was not considered in the paper of Sneddon
[10]. However, it is easily included [7, 8] in the formulae by
introducing, instead of the plane strain modulus E
∗
, a reduced
modulus E
r
that is defined as
1
E
r
=
1 − ν
2
i
E
i
+
1 − ν
2
s
E
s
, (11)
where ν is Poisson’s ratio and the subscripts i and s stand for
indenter and sample, respectively. It is possible to consider the
indenter displacement in this way because, for two bodies in
contact, there is a general relation for the displacements w in
the normal direction:
w
i
w
s
=
(1 − ν
2
i
) · E
s
(1 − ν
2
s
) · E
i
, (12)
which is valid for any arbitrary shaped body without sharp
edges (see [12]). For a given applied force, the contact area
of a deformable indenter is larger than that for a rigid one and
the contact stiffness decreases. It is important to note that this
formula cannot be applied to a perfect cone due to the point
3
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 2. Comparison of pressure profiles below the contact area
for different indenter geometries. The local pressure, p,is
normalized to the mean pressure, p
m
, under each geometry and the
radial distance, r, is normalized to the contact radius, a. The elastic
calculations were performed for equal contact radius.
of discontinuity at the tip. A solution for the cone can only
be obtained when the cone angle is allowed to change as a
function of the modulus ratio of the two contacting bodies. It
is not given here because it is not required in the following.
2.3. Determination of the elastic surface deformation when
there is additional plastic deformation
In section 2.2 we considered only wholly elastic deformation.
The question arises as to whether this analysis can also be
applied when there is an additional plastic deformation. To
provide an answer, consideration of the contact pressure for
elastic conditions and for different indenter geometries is
useful. In figure 2 the local contact pressure, p, normalized by
the mean contact pressure, p
m
, is plotted as a function of the
radial distance, r, normalized to the contact radius, a.
For a conical indenter the pressure becomes infinite
in the centre of the contact. The same is the case at
the edge of a flat punch. No material can resist such
a pressure singularity and plastic flow immediately arises,
which reduces the stresses significantly. The final (yielded)
pressure distribution (resulting from plastic deformation
having occurred) cannot easily be estimated analytically
but can be estimated numerically, e.g. by FE calculations.
Accurate FE calculations for pointed indenters (e.g. [13])
rarely give the contact pressure distribution at maximum force.
An early attempt by Bhattacharya and Nix [14] (figure 3)
compares the Hertzian pressure distribution with the pressure
distributions for plastic deformation of two materials with
different H/E ratios under a conical indenter. It can be seen
that the high pressure in the centre is drastically reduced and
that the pressure distribution for the harder material is similar
to a Hertzian pressure distribution. In contrast, the pressure
distribution in the softer material is nearly constant over the
large majority of the contact. Thus the pressure distribution
changes due to plastic deformation and so does the elastic
deformation and the ratio h
s
/h
max
, even though the indenter
geometry and material elastic modulus remain the same.
Figure 3. Comparison of the Hertzian pressure distribution with
contact pressure distribution at maximum force for plastic
deformation with a conical indenter.
An indication of the pressure profile after plastic
deformation can be obtained from an analysis of the force-
removal curve. Oliver and Pharr [7] have shown that the
force-removal curve from elasto-plastic deformations with a
Berkovich indenter can be described by a power function
F = C · (h − h
0
)
m
. (13)
The exponent, m, for a Berkovich geometry should be 2 for
an elastic contact, as given in equations (7a), (7b); however,
they obtained experimental values from force-removal curves
of 1.2 <m<1.6 and found that the best interpretation
of the experimental data was possible for an epsilon value of
0.75. This corresponds to a force-removal curve exponent of
m = 1.5, i.e. the value for a spherical indentation rather than
the expected value of 2 for a conical or pyramidal indenter. FE
calculations yield the same result. It is for this reason that the
current standard, ISO 14577, requires an epsilon value of 0.75
for indentations with Vickers and Berkovich indenters.
2.4. Improvements in the analysis of the elastic part of the
deformation
2.4.1. Variable epsilon factor. Experimental results for
plastic deformation, where the force-removal exponent, m,
is between 1.2 and 2, have shown that the exponent is not
a constant determined by the indenter geometry, but is a
material parameter which may differ between force-increasing
and force-removal. Neither the elastic solution for a conical
indenter nor that of a spherical indenter can describe the
situation properly. An improved model is the ‘equivalent tip
model’ of Bolshakov et al [15]. Here, the idea is that an
axisymmetric indenter of arbitrary shape, pressed against a flat
surface, can be defined such that it produces the same pressure
distribution as a conical indenter pressed against a previously
plastically indented surface. A solution from Segedin [16]
and Sneddon [10] can therefore be used to calculate the elastic
deformation caused by such an indenter. The shape of the
4
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
1.0 1.2 1.4 1.6 1.8 2.0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
h
s
/h
h
s
/h
m
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Cone
Sphere
ε
ε
Figure 4. Ratio of the elastic displacement outside the contact zone,
h
s
, to total elastic deformation, h, and epsilon factor, ε, as a function
of the force removal exponent, m.
‘equivalent tip’ can be described by the series
z =
∞
n=1
c
n
r
n
, (14)
where r is the radial co-ordinate and z the distance of the
surface from the tip. The penetration can be expressed as
h =
√
π
∞
n=1
n
2
+1
n
2
+
1
2
c
n
a
n
, (15)
where is the factorial or ‘gamma’ function and a is the
contact radius. This solution was taken by Woirgard and
Dargenton [17] and simplified. A great simplification can be
achieved if the possible geometries of the tip are restricted to
ones that can be described by a single term with a non-integer
exponent. In this case, the exponent of the shape function n is
correlated with the exponent, m, of the force removal curve by
the simple formula [17]
n =
1
m − 1
, (16)
and the ratio h
s
/h
max
is given by
h
s
h
max
= 1 −
1
√
π
m
2m − 2
2m − 1
2m − 2
. (17)
The result for h
s
/h
max
and ε as a function of m is given in
figure 4. In this way, the force-removal curve exponent can
provide some valuable information about the true pressure
distribution after the plastic deformation that occurred during
force increase. Oliver and Pharr came to the same conclusion in
their later paper of 2004 [18] where they conclude that ε = 0.75
is a reasonable estimate although one could easily rationalize
slightly different values.
Figure 5. Comparison of the normalized contact pressure for
indenters with different shape functions according to equation (14)
using only one term, n = 1, which is expressed by the force removal
curve exponent m.
Figure 5 shows the pressure distribution for exponents in
the measurable range between 1.2 and 1.6. It can be seen
that the main variation occurs at the centre of the contact.
However, the area of the central region as a proportion of
the total indentation contact area is relatively small. Thus,
even if the description of the real contact pressure in the
centre is inaccurate, it has a relatively low influence on the
calculation of the overall elastic displacement. Furthermore,
the elastic surface displacement, normalized to the contact
radius, depends only very weakly on the exponent m, i.e. h
s
/a
lies between 0.23 and 0.26 while h
c
/a varies much more. A
positive effect of this is that uncertainties in the determination
of m and the simplifications of the model do not have much
influence on h
s
. For indentations with pointed indenters, the
epsilon value itself can vary between 0.72 and 0.8.
Not all indentation users have access to software that
can easily calculate gamma functions. To assist these users,
we have found a more accessible approximation to the ε(m)
behaviour based on (10) and (17). This approximation has a
high accuracy in the range 1.05 m 5 (deviating from the
accurate solution by less than 0.5%) and is as follows:
ε(m) =
0.081 58
√
m − 0.94
−
0.616 79
(m − 0.94)
0.02
+
1.263 86
(m − 0.94)
0.001
.
(18)
Note that, even when an accurate value of the variable epsilon
factor based on (10) and (17) is used instead of a constant
value of 0.75, it is still an approximation, whose accuracy
depends on the material and the true pressure distribution after
plastic deformation. In particular, this model cannot correctly
describe a constant contact pressure, since this would require
both a linear force-removal curve and a contact radius that is
kept constant. However, this does not happen in practice, not
even at the very beginning of force removal.
The situation becomes worse for indentations into
coatings where there is a modulus and Poisson’s ratio mismatch
between coating and substrate. In this case, the variable
force-removal exponent approach can no longer give a good
5
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 6. Normalized elastic indentation depth of a conical indenter
h/a
0
over the normalized distance from the indentation centre r/a
0
.
The total displacement of distinct surface points is expressed by
dotted lines. The radial displacement at the contact edge is given
by u.
estimate of the h
s
/h ratio because it now also depends on
the ratio of substrate-to-coating properties (i.e. modulus and
Poisson ratio) and also the contact radius-to-film thickness
ratio. Generally, the accuracy improvement obtainable by
using a variable epsilon factor instead of a constant value
of 0.75 is relatively small. Nevertheless it may improve
the comparability of modulus results with those from other
methods, as will be shown later.
2.4.2. Radial displacement correction. Sneddon [10]gave
elastic solutions only for penetration in the normal direction.
He did not derive formulae for the complete elastic field.
Hanson, however, has done so for spherical [19] and conical
[20] indenters and Fabricant has done so for a flat punch
[21]. All these solutions include radial elastic displacements
parallel to the surface plane because they were derived by
using a specific pressure distribution as the boundary condition
rather than the shape of the indenter. It is interesting to note
that, although the pressure distribution was determined using
the initial indenter shape, the resulting surface deformation,
including the radial contribution, does not conform exactly to
the real indenter shape. The subtle consequences of this are
shown in figure 6 for a conical indenter using the results of
Hanson [20] and comparing them with the shape of a rigid
indenter indented to the same depth and our prediction of
the FEA results that would be obtained for the same contact
area. Equidistant points on the original (undeformed) surface
(shown as filled circles) are compressed under force in the
centre and strained at the contact edge. The co-ordinate system
for the analytical calculation is that of the undeformed surface;
yet this is distorted under force such that an atom at the contact
edge at a
0
is found at position a under force. It can be seen that
the resulting surface shape in the r–z-plane under the contact
area does not coincide with the surface of the indenter as it
should do, i.e. it is not a straight line as it should be for a rigid
cone.
It is important to understand the effect of this on the
different calculations performed in indentation analysis. An
indentation into a material of modulus E produces a contact
under force with stiffness, S, and radius a at a contact depth
h
c
. Sneddon and Hanson agree on the contact depth. Thus,
if the area function used in the analysis is taken to be the real
(unloaded) shape of the indenter, the value of h
c
(measured
from the tip at the centre of the indent) corresponds to a radius
a
0
which is larger than a by amount u. In contrast, if the
indenter area function is to be derived by the indentation of a
reference material, the effective contact area is calculated from
the stiffness equation. This will yield a value (at a contact
depth h
c
)ofπa
2
, which is less than the πa
2
0
value that would
be measured at the same depth by direct methods (e.g. AFM).
It is therefore important to be aware of this when comparing
results from different sources. Using FEA analysis to obtain
the radial co-ordinate of the last node in contact is a common
way of obtaining the radius of contact, but care needs to be
taken that the correct value of h
c
is used if this is to be assigned
to the creation of a virtual area function.
To calculate the value of u for elasto-plastic deformation,
we find it is preferable to use the radial displacement for
a spherical indenter since it is a good approximation to the
pressure distribution after plastic deformation for a wide
range of materials. A solution has already been given by
Johnson [22]:
u =
−(1 − 2ν)(1+ν)
2E
p
m
a
2
r
,r a, (19)
where ν is Poisson’s ratio, p
m
is the mean contact pressure,
a is the contact radius and r is the radial distance from the
centre of the indent. This formula is valid for any rotationally
symmetric indenter. Therefore, it makes no difference whether
a cone or a sphere is used if the contact radius is equal. The
minus sign denotes that the surface displacement u is in the
negative r direction (i.e. towards the indent centre). Since, by
definition, p
m
is identical to the hardness, H ,atr = a one
obtains
u
a
=
−(1 − 2ν)(1+ν)
2
H
E
. (20)
An additional correction that must be considered is that, after
plastic deformation, the surface is no longer flat and so the
elastic radial displacement must now be estimated along the
new deformed surface within the impression. This can be
calculated by the projection of u onto the original surface
position. This results in a correction term cos(α
r
), where α
r
is the residual angle between the face of the impression and
the original surface plane and is slightly smaller than the facet
angle, α, as shown in figure 1. α
r
is variable within the range
given by
α α
r
tan
−1
h
0
a
, (21)
where h
0
is the indent residual depth and α
r
is always below
20
◦
for ISO14577 compliant Berkovich or Vickers indenters;
the correction is, therefore, at most 6%. The final formula
is then
u
a
=
−(1 − 2ν)(1+ν)
2
H
E
cos(α
r
), (22)
and the radial displacement correction according to (22)is
largest when α
r
is smallest. Since u is directed inwards
6
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
under pressure, the contact radius increases by u during
force removal. The size of the residual indent is therefore
systematically underestimated by the calculation of A
c
from
the contact depth. The radial displacement correction is
proportional to the H/E ratio. It is, therefore, very small for
most metals (<0.5%) but reaches larger values (up to 5%) for
highly elastic materials such as fused silica. With a hardness
of about 9 GPa, a modulus of 72 GPa and Poisson’s ratio of
0.17, fused silica is one of the most elastic materials with a
value of H/E = 0.125. It is very often used as a calibration
material for the indenter area function yet, if the corrections
described here are not applied, it results in a systematic error
for all subsequent results obtained on materials with different
radial corrections (i.e. most materials other than fused silica
itself).
As in the discussion of figure 6, equation (22) implies that
the contact area at a given indentation depth in the real indenter
is larger than so far supposed or calculated. That means
that, if the real indenter area function is used, the calculated
hardness and modulus are smaller than the reference value for
the material. Conversely, for a given hardness the indentation
depth is larger than that calculated without correction.
2.5. Derivation of the complete force–displacement curve
If we take Segedin’s [16] superposition formula for a
rotationally symmetric punch of arbitrary shape limited to only
one term and equations (11) and (14), it follows that, for elastic
deformation:
F(h
e
) =
2
m
· E
r
· a(h
e
) · h
e
. (23)
For elasto-plastic deformation with a cone we can replace a(h
e
)
by substituting the hardness definition. Using (9) the elastic
surface deformation outside the contact area is
h
s
=
√
π · ε(m)
2
·
√
H
E
r
·
√
F. (24)
With (2) and (4) we now have both the elastic and plastic
parts of the deformation and can finally derive from (24) the
force-increasing curve of homogeneous and isotropic materials
deformed with an ideal cone:
F =
4 · π · E
2
r
· H
(2E
r
· tan α + π · ε(m) · H)
2
· h
2
. (25)
The quadratic force–depth dependence has already been
described by Hainsworth et al [23], but they used only
two empirical constants for the proportionality factors.
Malzbender and de Wirth [24] and Troyon [25] have derived
the same formula as (25), the only difference being that they
used a constant epsilon of 0.75 (Malzbender) and an additional
correction factor γ for radial displacement (Troyon). Further,
Malzbender tried to incorporate tip imperfections by adding a
depth offset and assuming that the quadratic depth dependence
remains. This, however, will not be the case, at least for
depths of the order of the tip radius because, unfortunately, the
derivation of (25) is limited to an ideal cone with a constant
hardness (contact pressure) and the latter assumptions are not
generally true in the elastic–plastic transition range. A more
detailed analysis follows below.
The absolute elastic deformation at F
max
is given, from
(3) and (23), by
h
e,max
=
m ·
√
π
2
·
√
H
E
r
·
F
max
. (26)
Describing the force-increasing curve by the power function
of equation (13), we obtain the proportionality factor C:
C =
F
max
m ·
√
π
2
·
√
H
E
r
·
F
max
m
. (27)
and the force-removal curve can be expressed by
F = C · (h + h
e,max
− h
max
)
m
for (h
max
− h
e, max
)<h<h
max
. (28)
This now allows the calculation of the complete force–
displacement curve given the assumption of depth independent
hardness and modulus.
Now equations (25)–(28) do not consider the radial
displacement correction. To include this correction one has to
go back to the substitution of the contact radius by the hardness
in (23). If we define the contact area as the projected area of
the indenter at the same contact depth but with the indenter is
under zero force (i.e. the area that would be directly measured
by metrological AFM or similar), then we must include the
radial correction to obtain the somewhat larger contact radius
for a given depth. Thus, a
= a + |u| and, using equation (20),
the radius is increased by the factor
1+K ·
H
E
; K =
(1 − 2ν) · (1+ν)
2
(29)
and the pressure is no longer H but H(a/a
)
2
, i.e. H is reduced
by the square of (29).
Equation (25) will then, to a first approximation,
transform to
F =
4 · π · E
2
r
H
1+K ·
H
E
2
2E
r
· tan α + π · ε(m) ·
H
1+K ·
H
E
2
2
· h
2
,
(30a)
which can also be written as
F = E
r
·
tan α
√
π
·
E
r
√
H
·
1+K ·
H
E
+
ε(m)
1+K ·
H
E
·
√
π
2
·
H
E
r
−2
· h
2
(30b)
7
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
to allow a direct comparison with the equations of Malzbender
and Troyon. (The additional angle dependent factor in (22)
is neglected here for simplification.) There is a single and
distinct difference between (30b) and the Malzbender and
Troyon equation. The correction factor for radial displacement
(29) appears as a scaling factor in the first term in the brackets
of (30b) but does not in the Malzbender and Troyon equations.
The physical meaning of this is that, in our methodology, the
correction is related to the area and not to the stiffness per se
and so is applied to both hardness and stiffness relationships.
It appears that, in the methodology of Malzbender and others,
the contact area under load is used without considering the
radial displacement and the correct relation to the stiffness is
then re-established by introducing a correction factor beta that
appears only in the stiffness equation.
Using equations (26)–(28) it is possible to directly derive
the force-removal curve stiffness for h = h
max
. A longer
calculation (not given here) results in
S =
4 · E
2
r
2 · E
r
· tan α + π · ε(m) · H
· h
max
. (31)
For a parabolic force-increasing curve and a homogeneous
material, the contact stiffness always shows a linear
dependence on the maximum penetration depth, independent
of the shape of the force-increasing curve. Considering the
radial displacement correction and assuming E
s
≈ E
r
(for
further simplification) the corrected contact stiffness is given
by the formula
S =
4 · E
r
· (E
r
+ K(ν) · H)
2
2 · tan α · (E
r
+ K(ν) · H)
2
+ π · ε(m) · H · E
r
)
· h
max
.
(32)
For example, the contact stiffness calculated for fused silica
without correction according to (31) (using H = 9GPa,
a force-removal curve exponent of 1.2 and a penetration
depth of 1 µm) is 268.4 mN µm
−1
. With radial displacement
correction, the contact stiffness at such a depth is slightly
higher, at 276.3 mN µm
−1
. Conversely, for the uncorrected
case, the predicted force to obtain a depth of 1 µm is 105.0 mN
while it is 101.0 mN after including the correction.
The description of the force-increasing curve immediately
places a limit on the generation of plastic deformation by a
cone. In the purely elastic case h
c
= 2h
e
/π and it follows
from (4) and (24) that
2
π
m ·
√
π
2
·
√
H
E
r
·
√
F =
tan α
√
π
·
√
F
√
H
. (33)
And substituting m = 2 (for a cone):
H
E
r
=
tan α
2
. (34)
For a Berkovich or a Vickers indenter with an equivalent
cone angle of 19.7
◦
it is therefore impossible to induce plastic
deformation in a material with H/E 0.179. It is therefore
not surprising that the Vickers hardness of rubber cannot be
measured. (In a Vickers hardness test, rubber appears to
have an infinite hardness, since no residual impression can
Table 2. Review of correction factors as a function of indenter
shape for the stiffness–area function relating to a conical indenter.
Indenter/ ββ
pressure (triangular shape) (quadrilateral shape)
Flat punch [28] 1.034 1.012
Pyramid [29] 1.141 1.051
Pyramid, FE 1.138 1.090
calculation [ 30]
Constant 1.0226 1.0055
pressure [31]
be produced.) Rubber has a low modulus resulting in an H/E
ratio above the 0.179 limit and plastic deformation in a material
with such a high H/E ratio is only possible if the exterior facet
angle is increased enough (i.e. the indenter is made sharper).
2.6. The beta factor
The contact stiffness can be obtained by differentiating
equations (7a), (7b). By suitable transformation, using the
known indenter geometry, one obtains in both cases the
universal formula for the relation between contact stiffness,
area and reduced modulus:
S =
dF
dh
=
2
√
π
·
A
c
· E
r
. (35)
Pharr et al [26] have shown that this universal relation is valid
for any axisymmetric indenter. Furthermore, deformation of
the material surface (for instance, by a hardness impression)
does not invalidate application of equation (34). Gao and
Wu [27] proved, by a perturbation method, that the contact
stiffness is insensitive to the cross sectional shape of a punch,
as long as the shape does not differ too much from a circle.
Pharr et al [26] introduced a correction factor, beta, on the
right-hand side of equation (35) to cope with deviations from
the ideal rotationally symmetric shape. This correction was
originally based on King’s [28] numerical calculations for
elastic deformation with a flat-ended punch. However, King
is not the only one who has made such calculations, see
table 2. Bilodeau [29] has performed a similar calculation
with an approximate solution for pyramids. Giannakopoulos
et al [30] used a very accurate FE calculation for the same
pyramidal geometries and Hendrix [31] has performed the
calculation for a constant pressure profile with triangular or
quadrilateral shape. All these calculations consider only purely
elastic deformation. As we know from the previous sections,
the infinite stresses produced by a flat punch (or a pyramid)
will be reduced by plastic deformation and a uniform contact
pressure is the most likely outcome. It is interesting to note
that a uniform pressure distribution assumption produces the
smallest corrections. Using a careful etching technique on
Vickers indents produced at high forces in steel, Weiler [32]
has shown that the elastic–plastic boundary is completely
rotational symmetric and that there is no preferred extension
in the direction of the diagonals. This means that the high
stresses at the edges, which were predicted by purely elastic
calculations, are greatly reduced by plastic deformation. For
these reasons, the standard, ISO 14577, does not use the beta
8
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
factor and it is indeed better to use beta = 1 for pyramidal
tips. In fact, there may be a slight deviation away from 1 for
highly elastic materials with a small plastic zone, but one can
expect that the error will be smaller than 5% in the worst case.
It is only when the deviation from the rotational symmetry
is very strong that a correction is required. Hendrix showed
this for a Knoop indenter [31], where he obtained a correction
factor of 2.68 for purely elastic deformation. For this reason, a
correction should be considered when using a Vickers indenter
that has a so-called ‘chisel edge’ at very low indentation depths.
In this case the contact area has lost rotational symmetry by
being much longer than it is wide.
In recent publications, for instance, by Oliver et al [18]or
Troyon [33], attempts have been made to consider the radial
displacement by using the beta factor. A correction factor
of more than 3% came out of FE calculations by Cheng and
Cheng [34] and Oliver and Pharr [18] if all corrections are
rolled up into the beta factor. The disadvantage of this method
is that it mixes geometrical corrections (i.e. dependence on
the indenter shape) with material dependent corrections (i.e.
dependence on the H/E ratio). Therefore, we consider it
to be better that any radial displacement correction of the
contact area be applied independently of a possible beta factor
accounting for indenter geometry.
In a recent publication, Strader et al [35] made an
experimental evaluation of the constant β relating the contact
stiffness to the contact area by using indents into fused silica
and analysing them with an accurate imaging technique. They
calculate the radial displacement by an FE method and get an
area increase of about 9.5%—very close to our FE result of
10% (see figure 8). The measured contact area after unloading
is corrected by this value so that they get a beta factor of
1.055 for a Berkovich indenter to correlate stiffness and area
according to equation (35). They first reduce the area by 9.5%
to find then that it is 11% too small (due to the square root
function we have to double the beta value of 5.5%). In our
methodology, we always use areas and area functions that
correspond to the real indenter shape in the undeformed state.
This is for several reasons.
• The area functions obtained by indentations into different
materials with well-known modulus can be compared
directly with each other and agree with the area function
obtained by direct scanning, for instance, with an AFM.
• Indentation hardness can easily be compared with
conventional Vickers hardness, which uses the indentation
area after unloading.
• The analytical equation (35), obtained for purely elastic
indentations, is only valid for the co-ordinate system of
the surface in the undeformed state. That is, the radial
elastic displacement under load is not included in A
c
in
equation (35) and so an extra correction is not necessary.
We find that the difference between both sides of equation (35)
amounts to 1.5%, which is well within the experimental and
the FE error limits. This confirms our estimation that a
beta factor (which is only intended to correct differences in
the stiffness between rotational symmetric and triangular or
quadratic shape) would be very close to one.
2.7. Energy considerations
The total energy expended during indentation (i.e. plastic
energy plus stored elastic energy that will be recovered upon
force removal) can be easily derived from equation (25)by
integration. The result is
W
t
=
4 · π
3
·
E
2
r
· H
(2E
r
· tan α + π · ε(m) · H)
2
· h
3
max
. (36)
The calculation of the elastic energy stored in the indented
material under force is obtained by integrating (13)or(28) and
substituting equations (26) and (27). The details are lengthy
and are not given here. The final result is.
W
e
=
4 · π
2
· m
m +1
·
E
2
r
· H
2
(2E
r
· tan α + π · ε(m) · H)
3
· h
3
max
. (37)
The energy ratio is then
η
IT
=
W
e
W
t
=
3 · π · m
m +1
·
H
(2E
r
· tan α + π · ε(m) · H)
. (38)
A more accurate analysis, considering also the lateral
displacement correction, can be obtained by scaling H by the
square of the factor in (29) and results in the formula
η
IT
=
3 · π · m
m +1
×
H · E
r
2 · tan α · (E
r
+ K(ν) · H)
2
+ π · ε(m) · H · E
r
)
.
(39)
For the above example of fused silica we get an energy ratio
of 0.64 without radial displacement correction and a ratio of
0.598 with correction.
The energy ratio is independent of the indentation depth
and depends mainly on hardness and modulus. Several authors
have already investigated the relationship between η
IT
and H
and E. On the basis of scaling relationships combined with
FE simulations Cheng et al [36] derived a linear dependence
η
IT
αH/E
∗
. Malzbender [37] compared the results of several
publications and concluded that there are larger differences
between the reported data and that the results do not converge
to one single relationship. This can be understood now, since
η
IT
also depends on the exponent of the force removal curve
and Poisson’s ratio. It should be noted that even (39)isto
some degree a simplification because, strictly, it is only valid
for ideal cones and materials with depth independent hardness.
Especially for real indenters with a tip rounding, equations (38)
and (39) can only be valid for indentation depths large in
relation to the tip radius. Experience further shows that it is
very difficult to obtain a material that does not show some depth
dependence of hardness over the first few hundred nanometres.
The energy ratio has the advantage that it does not depend
on the indentation depth and therefore not on pile-up or sink-
in effects. This can for instance be used for the calculation of
hardness if the modulus of the sample is known.
9
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
2.8. Alternative formulae for the calculation of hardness and
modulus
Equations (31) and (38) can be used for the derivation of
alternative formulae for hardness and modulus that do not
require knowledge about the dimension of the contact area
and the absolute value of the elastic surface deformation. The
force-removal curve stiffness S and the energy ratio η can
be determined directly from the force–displacement curves.
Using them one obtains after further calculations
H =
3 · m · (m +1) · (tan α)
2
π
×
S · η
((m +1) · ε(m) · η − 3 · m)
2
· h
max
, (40)
E
r
=
−3 · m · tan α
2
·
S
((m +1) · ε(m) · η − 3 · m) · h
max
.
(41)
Again neither formula contains the radial displacement
correction. A consideration of the correction would it make
impossible to solve the equations for E and H and only
complex numerical methods could be applied.
The use of equations (40) and (41) makes the relatively
complicated determination of the area function obsolete.
However, both formulae still have the disadvantage that they
cannot eliminate uncertainties due to pile-up effects. Pile-up is
in principle a shift of the surface position during the indentation
process that cannot be detected during the measurement. The
measured value h
max
does not contain the surface shift and
is therefore somewhat too small. On the other hand, creep
effects during the hold period of the maximum force will
not invalidate equations (40) and (41) once the additional
energy absorbed during the creep period has been considered
in the calculation of the total energy, because h
max
and S
are determined after creep. Note, however, that although the
equations are also valid after creep, the value of H obtained
still depends on the amount of creep that has occurred. H
reduces as the amount of creep increases and h
max
changes,
while E
r
should remain unaffected by creep in formula (41)if
the amount of pile-up is still proportional to h
max
. It is a direct
consequence of the definition of hardness as a mean pressure
that, for materials with significant creep, H values can only
be compared for indentation cycles allowing equal creep time,
whatever analysis is used.
In contrast to the energy-based method above, the two
slopes technique, proposed by Oliver [38], suffers from
creep effects. Oliver uses the slope of force-increasing and
force removal curve at maximum force for the calculation of
hardness and modulus without knowledge of the area function.
If creep occurs (and it is always activated by the onset of
plastic deformation), the slope of the force-increasing curve
corresponds to a different (higher) hardness value than that of
the force removal curve where creep has enabled the same force
to create a larger indentation depth. This makes the solution
inaccurate especially for rapidly creeping materials such as
soft metals.
Since an ideal pyramid was assumed in the derivation of
equations (40) and (41), these equations deliver relatively more
accurate results for larger indentation depths, where the tip
rounding does not play a role and the relationship Fαh
2
is
approximately valid for the force-increasing curve. They are,
therefore, not a real alternative for the calculation of hardness
and modulus in the nano-range, where the assumption of a
perfectly pyramidal tip shape is unrealistic and will introduce
considerable error.
3. Verification by FE calculations
FE calculations have been carried out as a verification of
the calculated contact pressure profile and the elastic radial
displacement. They are themselves validated by comparison
with experimental results. The ABAQUS standard package
was used to construct an axisymmetric FE model to simulate
the quasi-static indentation of a semi-infinite sample by a rigid
indenter. The sample was modelled using 6200 linear four-
node quadrilateral elements (CAX4 or CINAX4), using about
12800 degrees of freedom with no radial displacement allowed
on the axis of symmetry. A flat strip of elements 3.4 µm thick
was placed on top of a quarter-circle mesh of radius 7.5 µm,
and infinite elements were added to the circumference of the
circle and the edge of the strip to allow the displacements to
decay to zero in an appropriate manner at the boundary of the
model. The mesh of the sample was designed such that the
elements in direct contact with the indenter (those in the flat
strip) were smaller than those in the underlying circular region.
Typical element dimensions at the sample surface were 40 nm
× 150 nm. The indenter was modelled as a rigid surface.
The axisymmetric cone-equivalent of a Berkovich indenter
was modelled as a triangle of depth approximately 2.5 µm and
half-width of 7 µm (i.e. a half angle of 70.3
◦
), with a rounded
tip of radius 0.2 µm at the point of indentation. All contact
between indenter and sample was assumed to be frictionless.
In all cases, the indentation was simulated by controlling the
motion of the top surface of the indenter. The contact pressure
calculations have been done for the same displacement of 1 µm
and the radial displacement calculations have been done for the
same maximum force of 100 mN.
The materials indented were modelled as uniform
isotropic materials. The following materials were used.
1. Fused silica E = 73 GPa, ν = 0.16, Y = 7 GPa, elastic
perfectly plastic.
2. Tungsten E = 411 GPa, ν = 0.28, plastic data from
compression test.
3. Sapphire E = 405 GPa, ν = 0.235, purely elastic.
The FE results for the contact pressure at maximum force
are given in figure 7. The results of Bhattacharya and Nix
[14], given in figure 3 for slightly different materials, are
of lower resolution but match the same general shape of
the pressure distribution. (The pressure profile for a purely
elastic indentation with a rigid sphere into sapphire agrees
well (as expected) with the Hertzian pressure profile shown in
figure 3.) The pressure in the centre of the contact is slightly too
low, which may give an indication of the uncertainties of the FE
calculation. It is clear that there are some accuracy problems at
the contact edge and in the centre at r = 0; nevertheless it can
10
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
r/a
Figure 7. Contact pressure profile normalized by the mean contact
pressure, p
m
, for elasto-plastic indention by a cone with a tip radius
of 0.2 µm into fused silica and tungsten and pure elastic indentation
by a sphere into sapphire. All indentation depths were 1 µm.
Additionally the analytical solution of a force removal curve
exponent of 1.4 from figure 5 is given.
be seen that the contact pressure for the elasto-plastic conical
indentations is below that of the elastic sphere. The high elastic
stress in the centre of the cone is drastically reduced by plastic
deformation. The force–displacement curve for fused silica
obtained by the FE calculation gave a force-removal curve
exponent of 1.33. The contact pressure curve should therefore
be situated between the analytical results for the force-removal
curve exponents of 1.3 and 1.4, given in figure 5. This is
indeed the case; however, the minimum at r/a = 0 is not
observed. There are several reasons for this effect. Firstly,
the elastic perfectly plastic material model for fused silica is
much too simple, mainly due to a deficiency in reliable material
data. For instance, the known densification of this amorphous
material is not considered. A force-removal curve exponent of
1.23 was determined from experimental force–displacement
data against 1.35 in the FE calculation. This difference in the
force-removal curve exponent serves as a guide to how well
the FE results describe reality. Secondly, the deformation of
the indenter is not considered which should also reduce the
stress in the centre. Thirdly, the analytical model itself is a
simplification, as mentioned previously.
The second verification concerns the radial elastic
displacement. The results are shown in figure 8. The position
of the last node that has contact with the indenter at maximum
force (i.e. the node at the contact edge) is tracked for the
whole force increasing and decreasing cycle. Due to elastic
deformation, the surface element is dragged inwards (i.e. in
a negative direction). Only for tungsten does the outward
plastic deformation dominate the elastic deformation. During
the purely elastic force removal (the right-hand portion of the
curves) the radial position of the surface element increases until
it reaches its final position at F = 0. This position is always
larger than the contact radius at maximum force if plastic
deformation has occurred. The relative radial displacement
(the ratio u/a) in figure 8 is given by the difference between
the positions at F
max
and F = 0.
Figure 8. Relative radial displacement of the last surface node of
the FE mesh that has contact with the indenter at maximum force;
given for sapphire (infinite strength), fused silica and tungsten. The
right parts of the curves belong to the purely elastic force-removal
cycle. Force increasing and decreasing coincide for sapphire.
The radial displacement amounts to 5.1% of a contact
radius for fused silica but only to 0.2% for tungsten. The
fused silica result is close to the analytical result of 4.4%,
given below in table 3. The difference is mainly caused by
having neglected the deformation of the indenter itself in the
FE calculation. Radial elastic deformation will be reduced if
the indenter can also elastically deform. The results show that
the elastic radial displacement can reach very significant values
for highly elastic materials, such as fused silica, and should not
be neglected for an accurate analysis of hardness or modulus.
4. Experimental verification
4.1. Variable epsilon value and radial displacement
correction
Clearly independent validation is required to check the
consequences and the correctness of using a variable epsilon
factor and a radial displacement correction. This can be
obtained by comparing the indentation-derived measurement
of the area function of the indenter tip with an independent
direct tip measurement method that delivers the area function
with high accuracy. One such independent method is the
direct measurement of the tip shape by a metrological AFM
whose calibration can be traced to National Standards. Many
AFMs still use ‘open-loop’ scanning or do not have an
independent, linear height measurement. Such instruments
normally do not reach the required accuracy to derive the
A
c
(h
c
) function because they suffer from piezo-hysteresis,
creep and non-linearity, particularly in the height measurement
axis. The corrections described in this work have now
been tested in a number of laboratories. Indentation-
derived area functions from a wide range of materials have
been compared with metrological AFM performed at the
National Physical Laboratory [39] and in the DESIRED project
[40]. In this study, a metrological AFM at the Physikalisch
Technische Bundesanstalt in Braunschweig, Germany, with
11
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Table 3. Relative change in the calculated effective contact radius for a Berkovich tip using variable epsilon factor (a
1
), radial
displacement correction (a
2
) and both (a
1,2
). m is the force removal curve exponent and A
c
is the relative change in the contact area.
Material ma
1
(%) a
2
(%) a
1,2
(%) A
c
(%)
Fused silica 1.23 2.05 4.39 6.54 13.07
Sapphire 1.53 −0.13 1.53 1.40 2.80
BK7 glass 1.35 0.59 2.92 3.52 7.04
laser interferometers mounted on all three axes, was used.
The interferometers used have an uncertainty of about 1 nm.
The absolute uncertainty was determined by comparison with
two holographic grids whose periodicity was known with an
uncertainty of 0.1 nm. The maximum difference between
the period length measured by AFM and the reference value
was 0.11 nm. More details can be found in [41, 42]. Shape
measurements were carried out on a Berkovich indenter with
a tip radius of about 0.7 µm. The same tip was used
for indentation measurements on fused silica, BK7 glass
and sapphire as reference materials within the European
project INDICOAT [43]. The indentation measurements
were performed at the Technical University of Chemnitz
using a UMIS-2000 nanoindenter (CSIRO, Australia). Eleven
different maximum applied forces in the range between
0.3 and 500 mN were used. For every maximum applied
force, ten replicate indentation measurements were carried
out and averaged after careful zero point and thermal drift
corrections. Only the averaged, corrected curves were used for
further analysis. The instrument compliance was determined
to be a function of the applied force and in the range
0.2–0.22 nm mN
−1
. The force removal curve was corrected
for frame compliance and fitted between 98% and 40% of
F
max
using the power function of equation (13). The contact
stiffness at F
max
was obtained after an extrapolation of the
power function to 100% F
max
. This procedure was used to
reduce problems with residual creep and instrument control at
the initiation of force removal. The fit also returned a value
for the exponent m.
Young’s modulus and Poisson’s ratio reference values
used for fused silica, BK7 glass and sapphire(0001) single
crystal were (72.5 GPa, 0.17), (83 GPa, 0.21), (405 GPa,
0.235), respectively. The reference value for the sapphire
single crystal is the Hill average of the elastic constants given in
[44]. For the diamond indenter, E = 1140 GPa and ν = 0.07
have been used.
When the contact depth h
c
and the square root of the
contact area A
c
are calculated with equations (9) and (35)
using a constant epsilon value of 0.75, the result is as shown
in figure 9 where it is compared with the shape of an
ideal Berkovich tip and the result of the AFM area function
measurement. It is clear that the agreement of the area
functions, obtained with different reference materials, is not
perfect. Fused silica differs the most from the AFM area
function whilst sapphire differs the least. Note that it is better
to use the square root of the area as ordinate since the ideal
shape is then presented by a straight line with a slope of 4.95
and area differences over the whole depth range are more easily
seen. Tip rounding causes the curvature over the first 300 nm
of the depth range and then the curves straighten out into an
approximately parallel course for depths greater than 500 nm.
0.0 0.2 0.4 0.6 0.8 1.0
0
1
2
3
4
5
Fused silica
Sapphire (0001)
BK7 glass
Ideal shape
AFM area function
Square root (A
c
) (µm)
h
c
(µm)
Figure 9. Comparison of area functions obtained without additional
corrections for different reference materials plotted as the square
root of the contact area versus contact depth. The ideal shape of a
Berkovich pyramid is given by the dashed line.
Figure 10. Area functions obtained using fused silica and sapphire
as reference materials after a slight variation of the instrument
compliance. The shape of an ideal pyramid is given by the dashed
line.
A number of different factors could cause disagreement
between the curves, e.g. an incorrect instrument compliance
correction or inaccurate elastic constants for the reference
materials. These possibilities are easily investigated by
modifying the input values slightly. Figure 10 shows the
result of a variation of the instrument compliance. Here the
compliance was reduced until the fused silica area function
agreed with the high depth end of the AFM area function. This
required a compliance value of 0.31 nm µm
−1
. However, such
a compliance variation gives rise to a much larger change for
the sapphire area function, because this material has a higher
stiffness and is therefore much more sensitive to the instrument
12
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 11. Comparison of area functions from figure 9 after
application of the variable epsilon factor and radial displacement
corrections. The shape of an ideal pyramid is given by the dashed
line.
compliance correction. Even when the compliance correction
is altered to force a match at large depths, it is impossible to
reach an agreement over the whole depth range of the fused
silica area function. A similar effect is found if the elastic
constants are varied. An agreement with the whole AFM area
function was not possible for fused silica. Best results were
obtained for E = 68 GPa which is significantly below the
reference value.
If, however, the area functions are calculated using
the two corrections—a variable epsilon factor and a radial
displacement correction—and equation (22), the square root
of the contact area
√
A
c
becomes
A
c
=
A
c
·
1+
u
√
A
c
/π
. (42)
The calculation of h
c
, u and A
c
has to be done iteratively since
the correction u also depends on A
c
. A converged result was
obtained after 5 to 6 iterations.
The result of the corrections is shown in figure 11. The
agreement between all area functions is very good. Only
below 300 nm is there a slight difference for the glasses. It is
thought that this difference is caused by a densification effect
in the amorphous glass structure and a corresponding sink-in
behaviour. It was observed during other measurements that
the difference is reduced for sharper tips.
The size of the correction varies with the material.
The correction for fused silica is the largest because it has
the highest H/E ratio and the lowest force-removal curve
exponent (only 1.23). A comparison of the effect of the
different corrections is given in table 3. Strictly speaking,
the effect of both corrections is not simply the sum of
each individual correction, because a variable epsilon factor
also influences the result for the contact depth. However, this
influence is so small that it can only be seen for fused silica.
Although the corrections on their own are not large, the
effect on the calculated contact area and therefore on the
hardness is considerable. The effect is especially important if
fused silica is used as a reference material for the determination
Table 4. Mean difference between conventional Vickers hardness
and indentation hardness and between indentation modulus and
Young’s modulus as an average of 19 different bulk materials
without and with variable epsilon factor and radial displacement
correction.
Calculation Mean hardness Mean modulus
method difference (%) difference (%)
Without corrections 27.4 16.0
With corrections 9.2 5.8
of the indenter area function. The indenter area can be
underestimated by 13%. If this inaccurate area function
is applied to future fused silica measurements, this has no
effect. However, if the incorrect area function is applied to
a material with a much smaller correction, e.g. sapphire, then
the difference between the two corrections will be revealed. In
the example of sapphire the uncorrected contact area would be
underestimated by 10.3% (i.e. 13.07–2.8%) and so the hardness
would be overestimated by 10.3%. The reduced modulus is
obtained from the square root of the contact area under applied
force at the corresponding contact depth. The contact area is
similarly reduced under force as described by the corrections
(see equation (22)) and results in overestimates approximately
half the size of those already calculated for hardness.
The material dependent corrections of table 3 have
now to be multiplied by a possible geometry dependent
correction factor (the beta factor) that corrects for the
non-rotationally symmetric stress field around pyramid-like
Berkovich impressions. As was shown before, this factor is
very close to unity.
Further validation of the derived corrections was
obtained in collaboration with the Federal Institute of
Materials Research and Testing in Berlin, Germany. The
conventional Vickers hardness of 19 different bulk materials
(including metals, glasses, ceramics and semiconductors
as well as cemented carbide) was measured at a force
of 500 mN. The indentation hardness (using a Vickers
indenter) was also measured at 500 mN with three different
instruments: Nanoindenter XP (MTS, USA), UMIS-2000
(CSIRO, Australia) and Fischerscope H100 (Helmut Fischer
GmbH, Germany) and converted into a Vickers hardness
number by applying a simple scaling factor (HV = 0.094 545
H
IT
). Additionally the indentation modulus was compared
with Young’s modulus obtained using other methods. Details
of this investigation can be found in [45]. The area function
calibration was done with fused silica (partially with BK7
for the Fischerscope). Measurement data were analysed with
the original software of the instruments (without additional
corrections) and then with third party (instrument independent)
software using both corrections.
Table 4 shows the differences in hardness averaged over
all materials and instruments. Using the corrections the
percentage differences between the measuring methods could
be reduced by about a factor of 3. Residual differences can
be attributed to pile-up and sink-in effects, which cannot be
corrected by this method.
13
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 12. Comparison of measured and calculated force–
displacement data for fused silica with an ideal area function.
4.2. Force–displacement curves
It was shown in section 2 that, for homogeneous materials
with depth independent hardness and modulus, both the force
increasing and decreasing curves could be calculated using
modulus and hardness as the only parameters. Accuracy of
the simulation can be improved if the force-removal curve
exponent m is also used, although m can only be obtained
experimentally.
In figures 12–14, curves calculated according to
equations (25)–(28) are compared with measurement data from
fused silica and sapphire at a maximum force of 300 mN.
The measurements were made with a Nanoindenter XP (MTS,
USA) at the Swiss Federal Laboratories for Materials Testing
and Research (Thun, Switzerland). A Berkovich indenter
(so-called ACCU tip) with a tip radius less than 100 nm was
used. As a first approximation, an ideal tip shape will therefore
be assumed. The hardness and modulus were calculated using
the real area function and all corrections. A Young’s modulus
of 72.9 GPa was obtained assuming a Poisson’s ratio of 0.17.
The hardness result was 9.19 GPa. The results for sapphire
were 419.3 GPa for modulus and 21.0 GPa for hardness.
The alternative calculation method without the known
area function, equations (40), (41), gave for fused
silica E = 71.95 GPa, H = 9.12 GPa and for sapphire
E = 436.4GPa, H = 21.9 GPa. These results agree well
with each other and show that equations (40) and (41) can
be applied if the assumption of an ideal cone is justified.
However, analysis of measurements at lower forces shows
that the discrepancy increases with decreasing force. This is
because the assumption of a constant tip angle is no longer
true and the application of equations (40) and (41) produces
uncertain results.
Measured force–displacement curves for fused silica and
those calculated using an ideal area function are shown in
figure 12. It can be seen that the agreement is not very good
for two reasons. Firstly, the continued depth increase (creep)
during the 30s hold period at maximum force was not taken
into account; secondly, the ideal area function results in an
Figure 13. Comparison of measured and calculated
force–displacement data obtained using the real area function.
Depth independent hardness and modulus values were used for the
description of the whole curve.
overestimation of the depth. Note that the shape of the force-
removal curve is not correct if a force-removal curve exponent
of 1.5 (corresponding to an epsilon value of 0.75) is used.
The creep can easily be allowed for by using different
hardness values for the force application and the force-removal
curve. The depth increase during the hold period for fused
silica was 22 nm. Subtracting this corresponds to a hardness
number before the creep period of 9.49 GPa.
In this analysis it is better to express the real area function
by using a function of h(F ) instead of F (h) for the force-
increasing curve. The plastic depth according to equation (4)
can then be calculated, using an iterative procedure with the
known area function, by an inverse solution of
A(h
c
) = F/H. (43)
h
c
is stepwise increased starting at zero until the left side of (43)
equals the right side. The assumption for this procedure is that
the hardness is still a constant, independent of the modified
tip shape. The contribution of the elastic deformation h
s
is
obtained as before using equation (24). The result is shown in
figure 13. The agreement for two different materials is now
excellent over the whole curve, even though depth independent
values for hardness and modulus were used.
4.3. Residual deviations from the model
Differences between theoretical and experimental curves will
be more obvious for indents at smaller forces since the tip
rounding causes depth-dependent hardness values and the
model applied is no longer valid. However, although the
agreement is excellent in figure 13 a detailed analysis shows
that, in the typical force range of a nanoindenter (<500 mN),
the exponent of the force-increasing curve never reaches two.
This is shown in figure 14 for a Berkovich indenter with a
relatively large tip radius of about 0.75 µm. The exponent of
the force-increasing curve was calculated locally using only
the data points in the immediate neighbourhood of the depth
14
J. Phys. D: Appl. Phys. 41 (2008) 215407 T Chudoba and N M Jennett
Figure 14. Local exponent of the force-application curve for several
materials. The measurements were made with a rounded Berkovich
tip (radius about 0.75 µm) at forces between 100 mN (nickel) and
300 mN. The transition from a small exponent, close to 1.5 for a
sphere, to larger values can clearly be seen.
point. The exponent starts at about 1.5—a value that represents
the elastic deformation with a sphere. Values smaller than 1.5
are probably caused by the influence of oxide layers on top of
the metals. The relatively large tip rounding in this example
influences the slope of the force-increasing curve for much
larger depths than the radius of the tip although a constant
hardness can be expected in this depth range. Another reason
for the deviation from the expected exponent of two is creep
of the material during the force-application process and the
dependence on the rate of force increase. The experiments
have not been carried out with a constant strain rate. The strain
rate decreases with increasing force. This has the consequence
that the relative creep (the depth increase from creep divided
by the local depth) changes as well, which can affect the
exponent of the force-increasing curve. However, the influence
of the variable exponent of the force-increasing curve on the
agreement between measured and calculated curves is small
as can be seen in figure 13. Only very slight differences can
be observed close to the maximum force.
A more severe deviation from the model is caused by
pile-up or sink-in effects. In section 2.1, an ideal flat
surface was assumed for the derivation of the equations. This
assumption is definitely wrong for most metals. A detailed
analysis of the pile-up behaviour as a function of the E/H ratio
and the work hardening behaviour based on FE calculation
has been carried out by Bolshakov and Pharr [46]. They came
to the conclusion that pile-up can lead to an underestimation of
the contact area by as much as 60%. To the best knowledge of
the authors there exists at present no method that can derive the
amount of pile-up from the force–displacement curve alone.
The error in hardness or modulus therefore depends on the
work hardening behaviour and the H/E ratio of the material
and this cannot be prevented. A correction is only possible if
the amount of pile-up is measured by a profilometer or AFM or
if the modulus of the material is known. Fortunately, pile-up
does not exist (or is very small) for most of the ceramics,
glasses, semiconductors or hard coatings. Independent of the
pile-up effect, the measurement error will be reduced by the
given corrections in relation to the standard analysis method.
5. Conclusions
Instrumented indentation has improved significantly over
the years. Improved calibrations of force, displacement,
indenter area function and frame compliance have made
measurements highly reproducible. It is now possible to detect
inconsistencies in the contact mechanics used to describe
instrumented indentation. Refinements are necessary to
resolve a material dependent error that remains after area
function, zero point and frame compliance have all been
corrected. The most important corrections are the use of
a variable epsilon and a radial displacement correction. It
is shown that the Oliver and Pharr beta factor is a less
important correction and is rarely necessary once the other
two corrections are applied. Once the corrections are applied,
instrumented indentation data obtained from any and all
reference materials with negligible pile-up or sink-in effect
may be used to obtain the real indenter area function, i.e. one
which agrees with independent direct measurements such as
traceably calibrated metrological AFM.
The derived formulae can be used to calculate force–
displacement curves as a function of estimated values of
hardness and modulus. This enables better planning of
indentation experiments since the expected indentation depth
and the amount of elastic recovery can be given. Furthermore,
an alternative method of calculating hardness and modulus
without knowledge of the actual tip shape is given, which is
best applicable at higher forces.
Acknowledgments
The help of many persons who contributed to the experimental
results in this paper is gratefully acknowledged. This concerns
especially M Griepentrog and A D
¨
uck from the Federal
Institute of Materials Research and Testing (Berlin, Germany),
K Herrmann from the Physikalisch Technische Bundesanstalt
(Braunschweig, Germany), P Schwaller from the Swiss
Federal Laboratories for Materials Testing and Research
(Thun, Switzerland) and F Richter form the Technical
University of Chemnitz (Chemnitz, Germany). Similarly the
authors would like to thank Louise Wright from the National
Physical Laboratory (UK) for providing the Finite Element
Modelling data and Neil McCartney for fruitful discussions
and rigorous checking of the mathematics.
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