J. ^ECH et al.: CHARACTERIZATION OF STRUCTURAL MATERIALS BY SPHERICAL INDENTATION
695–698
CHARACTERIZATION OF STRUCTURAL MATERIALS BY
SPHERICAL INDENTATION
KARAKTERIZACIJA STRUKTURNIH MATERIALOV PRI
SFERI^NEM VTISKOVANJU
Jaroslav ^ech, Petr Hau{ild, Ondøej Kováøík
Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Department of Materials, Trojanova 13, 120 00,
Prague 2, Czech Republic
jaroslav.cech@fjfi.cvut.cz
Prejem rokopisa – received: 2016-10-19; sprejem za objavo – accepted for publication: 2017-01-16
doi:10.17222/mit.2016.302
Nano-indentation with spherical indenters is a method of mechanical properties characterization that could be used for the
determination of stress-strain curves. A necessary condition for an evaluation of the results is an exact knowledge of the indenter
shape. In this study, the shapes of two spherical indenters with a nominal radius of 20 μm were investigated by atomic force
microscopy and by indentation into materials with a known Young´s modulus. It was found that the actual indenter differs from
the ideal shape and it is affected by the crystallographic orientation of the diamond crystal. The effective radius determined from
the indentation measurements depends on the material deformation characteristics and it was lower than the actual radius. The
computed representative stress vs. representative strain curves of steel samples were significantly affected by the actual radii of
the investigated indenters.
Keywords: nano-indentation, spherical indenter, actual indenter shape, representative stress, representative strain curves
Nanoindentacija s sferi~nimi vtiskovalci je metoda karakterizacije mehanskih lastnosti, ki se lahko uporablja za ugotovitev
napetostno-deformacijske krivulje. Pogoj za oceno rezultatov je to~no dolo~eno poznavanje stanja vtiskovalca. V {tudiji sta bili
raziskovana dva sferi~na vtiskovalca z nominalnim radijem 20 ìm, ki sta bila raziskovana z AFM in z vtiskovanjem v material
z Youngovim modulom. Ugotovljeno je bilo, da se doti~ni vtiskovalec razlikuje od idealne oblike in, da nanj vpliva kristalo-
grafska orientacija diamantnega kristala. Efektivni radij, ugotovljen z meritvami vtiskovanja je odvisen od deformacije
karakteristik materiala in je bil ni`ji od dejanskega radija. Izra~unane reprezentativne napetostne krivulje jeklenih vzorcev so
bile ob~utno pod vplivom dejanskih radijev preiskovanih vtiskovalcev.
Klju~ne besede: nanoindentacija, sferi~ni vtiskovalec, dejanska oblika vtiskovalca, reprezentativna napetost, reprezentativna
napetostna krivulja
1 INTRODUCTION
Nano-indentation is an effective method for the char-
acterization of mechanical properties for very small vol-
umes of material. It is advantageously applied when
standard tests (e.g., tensile, fracture toughness tests) can-
not be used. It can be employed to characterize thin
films, welds or individual phases of a multi-phase mat-
erial.
Sharp indenters (such as the Berkovich three-sided
pyramid) are the most frequently used due to the simplic-
ity of the data interpretation. On the other hand, they do
not provide any information about the evolution of the
elastic and plastic stress-strain field under the indenter.
In spherical indentation, the stress and strain progres-
sively increase with the penetration depth and thus the
stress-strain curves of the materials can be determined.
Several models are used for a description of the
elastic-plastic stress-strain field under the indenter. At
the beginning of the indentation process, only elastic
deformation occurs and Hertz theory can be applied.1
With increasing penetration depth, the plastic zone starts
to evolve under the surface below the indenter and it
spreads to the surface.2 When the plastic zone is fully
developed, the ratio of hardness to yield strength
stabilizes at an approximate value H/
y = 3.3 This ratio
expresses the restriction of plastic deformation under the
indenter compared to the tensile tests. Several models
make it possible to evaluate stress-strain curves in whole
range of deformations.4 Methods determining the yield
strength, strain hardening exponent and other mechanical
characteristics do exist; however, several problems have
to be considered when analyzing the experimental data.
The crucial factor is a knowledge of actual indenter
shape.5–9 The real indenter does not match its ideal
shape. Imperfections are created already during the in-
denter’s fabrication and the indenter is progressively
worn with usage. The shape and properties of the in-
denter also depend on the crystallographic orientation of
the diamond crystal.10 The projected indenter area func-
tion Ap effectively corrects the imperfections of Berko-
vich, Vickers, or conical indenters. However, for the
spherical indenters, the actual projected area is not
frequently used as it cannot be directly related to the
actual radius, giving a tangent at the contact. Instead, the
effective radius of the indenter Reff can be used.
In this study, two diamond spherical indenters with
the nominal radius R = 20 μm were investigated. Their
Materiali in tehnologije / Materials and technology 51 (2017) 4, 695–698 695
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
UDK 620.1:67.017:620.3 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 51(4)695(2017)
actual shape was determined by direct (atomic force mi-
croscopy) and indirect (indentation into materials with a
known Young’s modulus) methods and its effect on the
stress-strain curves of structural materials was evaluated.
2 MATERIALS AND METHODS
Indentation tests were carried out on Anton Paar
CSM NHT nano-indentation instrument. Two steel sam-
ples (hardness etalon 487 HV from ball bearing carbon
steel and bainitic JRQ steel used in the nuclear industry)
were investigated by two spherical diamond indenters
(denoted as indenter A and indenter B) with nominal
radius R = 20 μm. Electrolytically polished samples were
loaded in the regime with partial unloading (CMC mode)
in the range of maximum force from 10 mN to 500 mN.
The whole indentation test consisted of 20 cycles with
progressively increasing maximum force. The increase to
the maximum force in the cycle takes 10 s, holding at
maximum force 5 s, and unloading to 20 % of maximum
force 10 s. Every unloading was analyzed using the
Oliver-Pharr procedure.11
Since the shape of the indenter is not supposed to be
ideal, the calibration procedure was performed for both
indenters. For a contact depth hc given by Equation (1):12
h h
P
Sc
= −max
max
.0 75 (1)
the projected contact area Ap was determined according
to Equation (2):12
A h
S
Ep c r
( ) =
π 2
24
(2)
where the maximum penetration depth hmax, maximum
applied force in the cycle Pmax, and contact stiffness S
are obtained from the force-displacement curve. Er
stands for the reduced modulus, comprising elastic pro-
perties of the indenter and the sample in Equation (3):12
1 1 12 2
E E E
i
ir
s
s
=
−
+
− 
(3)
The Young’s modulus and Poisson’s ratio of sample
Es and s were determined by independent ultrasound
pulse-echo measurement, and Ei and i are the known
Young’s modulus and Poisson’s ratio of the diamond in-
denter, respectively. The indenter effective radius can be
obtained from the geometry of the system (relation bet-
ween the projected contact area and the contact depth):
R
A
h
heff
p
c
c
=
+
π
2
2
(4)
Representative stress 
repr vs. representative strain repr
curves were determined using Tabor formulae in Equa-
tion (5):3

 repr =
P
C a
max
π 2
,  repr = 0 2.
a
R
(5)
where C is a constraint factor (C = 3), a is the contact
radius, and R is the radius of the indenter (nominal or
effective).
The tips of the spherical indenters were also exam-
ined by atomic force microscope AFM Park XE-100
with closed-loop z-piezo in tapping mode. An area of
20×20 μm2 was scanned with the resolution of 256×256
points. The data were corrected to the drift in slow scan-
ning axis and the noise was filtered. The projected area
Ap for the depths up to 2 μm was estimated from AFM
measurements as a surface of a planar cut through the in-
denter perpendicular to the load axis. From the measured
Ap(hc) dependency, the actual radius of the indenters was
computed using Equation (4).
3 RESULTS AND DISCUSSION
3.1 Atomic force microscopy
The topology of the indenters A and B and the maps
of their local radius of curvature obtained from the anal-
ysis of AFM data are shown in Figure 1. The local ra-
dius of curvature is defined as a radius of the sphere
interpolated in the circular neighborhood (1 μm in ra-
dius) of the particular point on the indenter surface.
Comparing the topology of both indenters, it can be seen
that the indenter B is more irregular than the indenter A.
The differences are also visible on the maps of local
radius of curvature, where the local radius of curvature
of indenter A at the apex is higher than for indenter B
and it reaches a higher value than the stated 20 μm.
For the data analysis, and also for the fabrication of
the diamond indenters, it is very important to know its
crystallographic orientation. The investigated indenters
were produced with the loading axis parallel to the direc-
tion [100] of the diamond crystal. The crystallographic
orientation has a significant effect on the indenter actual
J. ^ECH et al.: CHARACTERIZATION OF STRUCTURAL MATERIALS BY SPHERICAL INDENTATION
696 Materiali in tehnologije / Materials and technology 51 (2017) 4, 695–698
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 1: AFM images of: a) the topology of indenters A and b) in-
denter B, c) local radius of curvature of indenters A and d) B with
shown axes of the four-fold symmetry (dashed lines)
shape and it can introduce deviation from its nominal
spherical shape. As the diamond crystal is anisotropic, it
is worn and grinded more easily in the softer directions.
The four-fold symmetry of the diamond crystal is clearly
visible in the figures of the local radius of curvature
(Figure 1c and 1d).
The actual radii estimated from Equation (4) are
shown in Figure 2. The radius determined by AFM de-
creases with depth for both investigated indenters. The
decrease is more abrupt for indenter A (from approxi-
mately R = 30 μm to R = 22 μm at a depth h = 100 nm
and h = 500 nm, respectively). For greater depths, the
radius of indenter A stabilizes at a nominal value of
R = 20 μm. The radius of indenter B decreases only
slightly from R = 24 μm at a depth 200 nm to R = 23 μm
at a depth 500 nm. For depths h > 400 nm it is higher
than the radius of indenter A and it reaches a constant
value of nearly 23 μm, far more than the nominal value
of 20 μm.
3.2 Indentation
Evolution of the effective radius determined from in-
dentation measurements on steel samples at low penetra-
tion depths shows a different trend compared with AFM
results. The effective radius increases from a very low
value of about 10 μm and it reaches its maximum at a
depth approximately h = 200 nm. These low values of ef-
fective radius result from the uneven contact between the
indenter and the surface of the sample. After electrolytic
polishing, the surface of steel sample is wavy, as sche-
matically shown in Figure 3. The indenter touches the
local surface asperities and thus the actual contact area
for a given penetration depth is not spherical, but it is ir-
regular and smaller than the theoretical one. According
to Equation (4) the final effective radius is subsequently
smaller than the actual radius of the indenter.
The maximum effective radius from the indentation
measurements differs for both investigated indenters and
materials. Higher values were obtained by indenter A
than by indenter B, similar to the AFM measurements.
For greater depths the evolution of the effective radius
follows the trend of the AFM measurements. The radius
of indenter A decreases with depth and it is lower than
the radius of indenter B from depths of about 600 nm.
The effective radius of indenter B is rather constant.
However, the effective radii could not be compared for
higher penetration depths since high hardness of etalon
487 HV and the maximum possible applied force
(500 mN) limit the maximum penetration depth into the
487 HV sample to approximately 1 μm.
The differences in the effective radius estimated from
indentation into JRQ steel and etalon 487 HV could be
explained by the difference of their elastic-plastic prop-
erties. It is well known that the contact area (which is
used for a determination of effective radius) depends on
the actual indenter shape and the material deformation
characteristics.14 The effects of pile-up or sink-in can in-
troduce the error in the determination of contact area by
as much as 60 %.15 The effective radii measured by in-
dentations on investigated samples are lower than the ra-
dii from the AFM. More significant difference is for JRQ
steel, which is probably caused by higher strain harden-
ing exponent.
Representative stress vs. representative strain curves
were evaluated according to Equation (5) and they are
presented in Figure 4. Each material was investigated by
both indenters. The results considering nominal radius
R = 20 μm and effective radius Reff obtained from the in-
dentation into the corresponding material were com-
pared. To avoid the problems with the transition from the
elastic to the plastic region and the errors arising from
the uncertainty of geometry of contact at low penetration
depths, the cycles with low maximum load were ex-
cluded from the analysis.
As the nominal radius is higher than the effective ra-
dius, flow curves of JRQ steel using R = 20 μm are lower
than the curves obtained with Reff (Figure 4a). The
curves based on nominal radius significantly differ for
both indenters, especially for lower deformations (i.e.,
low penetration depths). Moreover, stress obtained with
indenter A using nominal radius decreases with increas-
ing deformation, which should not occur as the material
is hardening. It is caused by an important difference be-
tween the nominal and the actual shape of indenter A at
low depths (Figure 2). Using the effective radius, the
flow curves obtained by both indenters are nearly equiv-
alent. Small differences could result from using Equation
(5) for evaluating representative deformation. The for-
J. ^ECH et al.: CHARACTERIZATION OF STRUCTURAL MATERIALS BY SPHERICAL INDENTATION
Materiali in tehnologije / Materials and technology 51 (2017) 4, 695–698 697
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 2: Radius of spherical indenters with nominal radius
R = 20 μm obtained by AFM and indentation measurements
Figure 3: Schematic sketch of contact between spherical indenter and
surface asperities13
mula for representative deformation is based on the ratio
of contact and indenter radii a/R, which is valid for ide-
ally spherical indenters. Actual deformation is given by
the contact angle between the indenter and the sample
surface. As the shape of the indenter is not ideally spher-
ical, approximation of this angle by a/R is not exactly
valid and it could introduce small errors into the values
of the representative deformation.
The decrease of the stress with strain was also ob-
served for hardness etalon 487 HV using indenter A and
R = 20 μm. Nevertheless, the differences in flow curves,
which results from using nominal and effective radius,
are less significant than for JRQ steel. For indenter B,
representative stress vs. representative strain curves are
nearly identical as the effective radius for higher penetra-
tion depths reaches approximately its nominal value
R = 20 μm.
4 CONCLUSIONS
The shapes of two diamond spherical indenters with
nominal radius R = 20 μm were investigated in this study.
Their actual and effective radius was determined by
AFM measurements and indentation into materials with
a known Young’s modulus, respectively. It was found
that the indenter shape is affected by the crystallographic
orientation of the diamond crystal. For low depths, both
investigated indenters are flatter (larger radius) than
stated by the supplier. For greater depths, indenter A
reaches the nominal value, and the indenter B has a
slightly larger radius. The effective radius measured by
indentation into JRQ steel and hardness etalon 487 HV is
lower than the actual radii, which is caused by the sur-
face irregularities (for lower depths) and materials defor-
mation characteristics (for greater depths).
Determined representative stress vs. representative
strain curves of JRQ steel and hardness etalon 487 HV
are affected by the used radius (nominal or effective). To
obtain indenter-independent results, effective radii deter-
mined for a given material should be used.
Acknowledgment
Support from Czech Science Foundation (project no.
GB14-36566G) and Czech Technical University in
Prague (project no. SGS16/172/OHK4/2T/14) is grate-
fully acknowledged.
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J. ^ECH et al.: CHARACTERIZATION OF STRUCTURAL MATERIALS BY SPHERICAL INDENTATION
698 Materiali in tehnologije / Materials and technology 51 (2017) 4, 695–698
MATERIALI IN TEHNOLOGIJE/MATERIALS AND TECHNOLOGY (1967–2017) – 50 LET/50 YEARS
Figure 4: Representative stress vs. representative strain curves ob-
tained using Tabor formulae: a) JRQ steel, b) hardness etalon 487 HV