Full text of DOI:10.1557/JMR.2002.0388

Depth-sensing indentation at macroscopic dimensions
Jeremy Thurna) Dylan J. Morris, and Robert F. Cook
Department of Chemical Engineering and Materials Science, University of Minnesota,
Minneapolis, Minnesota 55455
(Received 4 June 2002; accepted 27 July 2002)
A macroscopic-scale depth-sensing indentation apparatus with the ability to be
mounted on an inverted microscope for in situ observation of contact events was
calibrated using the Oliver and Pharr [J. Mater. Res. 7, 1564 (1992)] procedure with a
two-parameter area function. The calibrated Vickers tip was used to determine the
projected contact area at peak load and the modulus and hardness of a variety of
non-metallic materials through deconvolution of the measured load-displacement
traces. The predicted contact area was found to be identical to the measured
area of residual contact impressions. Furthermore, for transparent ceramic materials the
projected contact area during loading was found to be the same as the area measured
from the diagonal of post-indentation residual contact impressions. The modulus and
hardness values deconvoluted from the load–displacement traces were compared with
independent measurements. The effects of sample clamping, column compliance, and
tip radius on the load–displacement data and inferred materials properties were also
examined. It is suggested that the simplicity of instrumentation and operation,
combined with the ability to observe indentations optically, even in situ, makes
macroscopic-scale depth-sensing indentation ideal for fundamental studies of contact
mechanics.
I. INTRODUCTION
A. Motivation
and analysis procedures of Oliver and Pharr are used to
examine a variety of nonmetallic materials with a two-
parameter area function introduced previously.8 The in-
dentation impression area, elastic modulus, and hardness
are determined from the P-h traces, and the areas are
compared with those determined from direct optical
measurements of the residual indentation impression.
The post-indentation corner-to-corner area is shown to be
equivalent to the contacted indentation area at peak load
for three transparent ceramics by mounting the indenter
on an inverted microscope and observing the contact
event in situ. Further, the effects of column compliance,
clamping of the sample, and indenter tip radius on the
P-h traces and the properties extracted from them are
examined. The loading traces are examined for applica-
bility of geometric similarity, and the unloading traces
are examined for applicability of the Oliver and Pharr
power-law analysis.
DSI at macroscopic length scales has been performed
on many custom-built2,9–20 and commercial devices
(such as the Instron MicroTester, Instron Corp., Canton,
MA, and the Zwick ZHU, Zwick GmbH & Co., Ulm,
Germany) and has numerous advantages. The most ob-
vious is the ability to observe the indentation contact area
and any related cracking directly, both in situ and post-
indentation, by optical microscopy.11 This will be taken
Depth-sensing indentation (DSI) has in recent years
become a common technique for characterization of met-
als,1 ceramics,2 and polymers.3 In particular, small-load
indentations are being used to characterize thin film and
surface layer properties4 and small-scale deformation
phenomena.5 Analysis of the load–displacement (P-h)
data recorded continuously throughout the load–unload
cycle provides information about the projected contact
area at peak load, which is coupled with the measured
unloading stiffness to provide estimates of the elastic
modulus and hardness. The analysis method, however,
requires careful calibration to determine the column
compliance and indenter tip shape. The most popular
calibration technique is that of Oliver and Pharr,6 which
is based on the elastic solution of Sneddon7 for indenta-
tion by an axisymmetric body. It is the purpose of this
work to examine this calibration technique at large loads
and macroscopic length scales at which the indentation
impressions can be examined optically. The calibration
a)Present address: Seagate Technology, Bloomington, Minne-
sota 55435
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
advantage of in this work in an effort to validate the
the contact displacement hc, and included angle of the
indenter 2␣. For conical indenters, A ס ␲a2 and for
Vickers indenters A ס 2a2. For metals and most ceramic
materials, the indentation impression diagonals measured
following the indentation event are assumed to be the
same as those at peak load, as inferred from experimental
investigations of Vickers indentation (both diagonals),23
Knoop indentation (the long diagonal only),24 and roller
indentation with “blunt” rollers (the track width).25 The
projected contact area at peak load can thus be related to
the indentation contact depth by simple geometrical ar-
commonly used deconvolution technique of Oliver and
Pharr6 to determine the impression area from the P-h
trace. The ability to observe crack initiation and growth
during the indentation cycle at loads typical for indenta-
tion toughness measurements was taken advantage of in
a similar instrument by Cook and Pharr13 to provide
valuable insight into the fracture behavior of ceramic
materials. DSI at these loads can also be used to provide
an estimate of the displaced volume driving crack growth
during unloading, a parameter that is lumped into a
material-independent constant in the relationship for in-
dentation toughness21 and indentation delamination
models for thick films.11,22 Simultaneously, DSI at mac-
roscopic length scales can be used to determine the
modulus and hardness of the materials for use in the same
models. In fact, it can be used to characterize the modu-
lus and hardness of thick films and coatings in much the
same manner that nanoindentation is commonly used to
examine thin films. The time-dependent properties of
soft viscoelastic and viscoelastoplastic materials and
coatings can be probed with relaxation tests as most mac-
roscopic DSI devices operate under direct displacement-
control (i.e., there are no feedback loops that complicate
creep and relaxation tests on commercial instruments).
Direct control over the displacement rate also makes the
device easier in general to control and build, while
the macroscopic scale of the device makes it easier and
cheaper to build. Similarly, the indenter tips are much
larger than for nanoindentation and are significantly less
expensive to purchase in a variety of shapes. While na-
noindentation is incredibly important for deducing the
elastic and plastic properties of thin films, macroscopic-
scale DSI is more advantageous for the study of the
mechanics of contact (especially during the contact
event) and the fracture and delamination resulting from
the indentation event.
2
guments. For a conical indenter, A ס ␲ tan2 ␣ hc .
Love’s elastic solution of conical indentation provided
relationships between the total and contact displacements
and between the load and contact displacement:26
h ס ␤hc and P ס AE* cot ␣/2, where E* is the plane–
strain modulus E* ס E/(1 − ␯2), E is the Young’s
B. P-h relationships during loading
and unloading
FIG. 1. Representative indentation load–displacement P-h trace of a
typical ceramic (in this case, soda-lime glass). Parameters used in
deconvoluting P-h traces to obtain material properties are indicated.
Indentation hardness H has been defined as
Pmax
H =
,
(1)
A
where Pmax is the peak indentation load and A is the
projected contact area at peak load. Figure 1 is a repre-
sentative load–displacement trace (on soda-lime glass)
showing Pmax and other variables used in this analysis.
Figure 2 is a schematic cross-section of the indentation
process showing the contact dimension a (a is the contact
radius for conical indentation and half the contact diago-
nal for Vickers indentation), the total displacement h,
FIG. 2. Schematic cross section of the indentation process during
loading of a ceramic material.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
modulus, ␯ is Poisson’s ratio, and ␤ ס ␲/2. These two
expressions can be combined to yield the P-h relation-
ship of a conical indenter:
where, k, hf, and n are fitted parameters (but on average
n should be around 3/2). The stiffness is then obtained by
analytic differentiation of Eq. (5). In the analysis here,
Eq. (5) will be fit to some portion of the unloading curve
to determine S.
The projected contact area at peak load A can be de-
termined from the residual impression if the diagonals do
not recover on unloading. It can also be determined from
the P-h trace using the procedure outlined in Oliver and
Pharr.6 First the contact displacement is determined from
the displacement at peak load hmax (Fig. 1):
␲ tan ␣
2␤2
P =
E*h2
.
(2)
The elastic solution for indentation by a four-sided py-
ramidal (Vickers) indentation is identical to Eq. (2), ex-
cept ␤ ס 1.45.27 Plastic deformation during loading
renders Eq. (2) invalid for most materials (except at ex-
ceptionally low loads), although P ∼ h2 for geometrically
similar indenters.28 Furthermore, Eq. (2) does not de-
scribe the P-h trace during a typical experiment because
all indenter tips have some finite tip radius R and perhaps
other imperfections that affect both the value of ␣ and the
quadratic relationship with contact depth shown above.5
In the absence of other, external, “independent” infor-
mation (such as the yield stress) the unloading slope is
the only unambiguous measure of the elastic contact
stiffness S ס dP/dh (Fig. 1). The compliance C סS−1 is
related to the elastic modulus of the material being con-
tacted by the following expression:7
hc ס hmax − ⑀Pmax
C
,
(6)
where ⑀ is a constant related to the form of the unloading
trace, here taken to be 0.75 (corresponding to n ס 3/2 for
a paraboloid of revolution). Ideally, this contact displace-
ment can be directly related to the projected contact area
of the indenter tip purely by geometry. Due to the afore-
mentioned imperfections in typical indenter tips, how-
ever, it is common to relate the area to the contact depth
by some area function A ס f(hc). There are many pro-
posed forms of f in the nanoindentation literature with a
variety of forms and with varying number of constants.
Some are based on geometrical arguments,8,33,34 others
on experimental measurements of the tip radius,35,36
while still others are almost completely empirical.6,37,38
The area function of Thurn and Cook is used in this
analysis because it specifically prescribes an effective tip
radius Reff and effective equivalent cone angle ␣eff based
on the calibration results and has the following form:8
1 2
ր
␲
E* =
.
(3)
ր
2
2CA1
Equation (3) is valid for indentation by an arbitrary axi-
symmetric indenter29 regardless of the strain hardening
properties of the material being indented or the presence
of residual stresses.30 The plane–strain modulus in
Eq. (3) is often replaced by the so-called “reduced modu-
lus” to account for elastic deformation of the indenter tip,
where the reduced modulus is defined as31
A = ␲ tan2 ␣eff hc + 4Reff ␲ hc
2
+ 4Reff ␲ cot2 ␣eff
.
(7)
2
An estimate of Reff is important for understanding the
shape of the P-h traces, especially at displacements on
the order of Reff. The parameter chosen here to repre-
sent the “lost” indentation volume due to tip truncation
Reff is also important for estimating the stress field and
understanding the shape of the P-h traces, especially at
displacements on the order of the tip truncation. The cone
angle is important for estimating the stress field around
the indenter and thus for comparing experimental re-
sults with finite element simulations and their resulting
scaling laws.39
2
2
−1
1 − ␯ ͒
1 − ␯ ͒
͑
͑
i
E =
+
,
(4)
ͩ
ͪ
r
E
Ei
where Ei and ␯ are the elastic modulus and Poisson’s
i
ratio of the indenter, respectively (here taken to be
1141 GPa and 0.07 for diamond).6 Equation (3) does not
provide any information concerning the shape of the un-
loading curve, so the method used to determine dP/dh at
peak load is a matter of some debate. If a specific type of
curve is fit to some portion of the unloading data (such as
a power-law curve), that curve can be related to the type
of indenter. For instance, Doerner and Nix32 used a
straight-line fit to the upper 33% of the unloading curve,
implying that the contact area did not change on initial
unloading (similar to a flat punch). Oliver and Pharr6
suggested a power-law curve with an exponent of 3/2,
implying that the indenter tip was best approximated by
a paraboloid of revolution:
II. EXPERIMENTAL
The indentation apparatus used in this analysis is de-
picted schematically in Fig. 3. Load was transmitted to
the sample via a sharp four-sided pyramidal (Vickers)
diamond indenter tip located at the end of a loading shaft.
The shaft was driven by a direct current (dc) servo mi-
crometer so the indenter tip was nominally under dis-
placement control. Displacements were imposed at rates
P ס k(h − hf)n
,
(5)
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
between 0.1 and 1 ␮m s−1, although for stiff materials
mode, the sample was clamped to the frame by gluing it
to an aluminum sled that was then bolted to the floor of
the indenter. The indentation apparatus is essentially the
same machine described in Cook and Pharr13 and used by
Suresh et al.18 As mentioned in Sec. I, many similar
devices have been built privately and commercially to
probe the elastic, plastic, viscous, and fracture properties
of various materials (Refs. 2 and 9–20 allude to some of
these devices). The nonmetal materials used in this
analysis were chosen to cover a wide range of mechani-
cal properties. Table I lists the materials examined with a
short description of each.40–51 All samples were polished
to a mirror finish (1-␮m diamond paste). The soda-lime
glass, sapphire, and NaCl were all optically trans-
parent and used for the in situ indentation observation
experiments.
such as those examined here, the imposed displacement
rate was significantly greater than the displacement rate
of the tip. Load was measured with a 100-N resistance
load cell in series with the indenter tip and displacement
measured with capacitance gauges at two points to elimi-
nate the effects of tilt in the loading shaft. The load and
displacement resolutions were 0.05 N and 0.05 ␮m,
respectively. The apparatus was designed to function in
two separate modes: the first involved mounting the
indenter atop an inverted microscope for in situ observa-
tion of the indentation event (on transparent materials)
and the second involved clamping the sample to the base
of the indenter (not shown) for precise displacement
measurements during the contact event. In the second
III. RESULTS
A. Sample clamping and column compliance
Macroscopic-scale DSI experiments were performed
previously, but the effects of column compliance and
sample clamping were sometimes ignored.2,12 The effect
of sample clamping can be seen in the P-h traces of Fig. 4
(on fused silica). The unclamped sample required con-
siderably more displacement to support a given a load.
Though not shown here, the moduli and hardness deter-
mined from the P-h curves of Fig. 4 using the Oliver and
Pharr method outlined below differed by a factor of two
(i.e., the unclamped sample showed values of modulus
and hardness half the magnitude of the clamped and lit-
erature values).
The compliance of column Cf was determined by treat-
ing the indenter and sample as springs in series so that
Ct ס Cf + C
,
(8)
where C ס S−1 is the sample compliance and Ct is the
total measured compliance. The sample compliance was
related to the projected contact area at peak load using
FIG. 3. Schematic diagram of the apparatus used here to perform
depth-sensing indentation at macroscopic loads.
TABLE I. Summary of materials used in this study.
Material
Description
E (GPa)
H (GPa)
Reference(s)
Soda-lime glass
Sapphire
Silicon
Al2O3–TiC
BaTiO3
Synroc
Polycrystalline alumina
Y-stabilized tetragonal zirconia polycrystal
Commercial microscope slide
99.995% Al2O3; (0001) single crystal
(100) single crystal
64/35 wt%, hot isostatically pressed
Polycrystalline, capacitor grade
Polycrystalline, titanate ceramic
Surgical grade alumina
70
425a
169
420
130b
189
400
220b
165
72
5.9
21.8
9.6
23.0
5.9
10.3
16.1
17.8
7.6
40, 13
41, 13
42, 2
43, 44
45, 46
46
46
47, 13
48
Annealed at 1300 °C
La2O3–Y2O3
Fused silica
NaCl
9.1 mol% La2O3 polycrystalline
Optically flat substrate material
Commercial FTIR window, single crystal
6.3
0.20
49, 13
50, 51
50b
ac axis.
bVoigt average from published elastic constants.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
Eq. (3) and a plot of Ct versus A−1/2 constructed from
corner-to-corner area of the residual impressions for each
indentation. The ordinate-intercept of a straight-line fit to
the data in Fig. 6 provided an estimate of the column
compliance Cf ס 0.021 ␮m N−1. Clearly this is not a
negligible contribution to the measured displacement, es-
pecially for large-load indentations on stiff materials
such as alumina. During tip calibration, the column com-
pliance is accounted for by replacing hmax with
hmax − PmaxCf and C with C − Cf in Eqs. (3) and (6).
Following calibration, it is simplest to subtract the PCf
product from the measured displacements prior to analy-
sis of the P-h traces.
large indentations on soda-lime glass. Figure 5 shows
P-h traces of two large indentations on soda-lime glass.
The unloading slope S increases with increasing load, so
C decreases with increasing load. The column compli-
ance was obtained from Fig. 6, showing the total meas-
ured compliance (from P-h traces such as those shown
in Fig. 5) against A−1/2. The projected contact area A
was determined by directly measuring optically the
B. The projected contact area and the indenter
area function
The abscissa of Fig. 6 was determined post-
indentation by measurement of the residual indentation
impression areas on soda-lime glass. In fact, this method
of determining A will be used extensively in this analysis,
so it is fitting to provide some experimental evidence for
the implicit assumption being made here, that the impres-
sion diagonal 2a does not significantly recover on un-
loading. It is well known that the depth of the indentation
impression recovers significantly on unloading.23 It
would seem plausible that the surface of the indentation,
too, might elastically recover on unloading. The amount
of elastic recovery of the diagonals during unloading on
soda-lime glass, sapphire, and NaCl was examined by
comparing the diagonal lengths measured during loading
with those measured from the residual impression.
Figure 7 shows P-h traces for indentations on all three
FIG. 4. P-h traces of fused silica with and without sample clamping.
(Neither P-h trace has been corrected for column compliance.)
FIG. 5. P-h traces of large indentations on soda-lime glass showing
the increase in contact stiffness S with increasing load P. (Both P-h
traces have been corrected for column compliance.)
FIG. 6. Column compliance calibration plot: total measured compli-
ance from the P-h data and projected contact area from measurements
of the post-indentation residual impression (soda-lime glass).
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
materials. Indentations were performed over a wide
with time. The unloading curve is nearly linear and
hf ∼ hmax, indicating little elastic recovery in the depth of
the impression on unloading.
range of loads on soda-lime glass [Fig. 7(a)] and sapphire
[Fig. 7(b)]. The loading curves lie on top of one another
and both materials show significant elastic recovery on
unloading (hf < hmax), as proposed above. Figure 7(c)
shows multiple indentations on NaCl to peak loads near
4.5 N. For these indentations, the indenter was held at
the peak imposed displacement for 30 s, resulting in the
small amount of creep and relaxation seen near the peak
where the load decreased and displacement increased
Figure 8 compares the contact area during loading and
the residual impression on these three materials.
Figure 8(a) is an image of the contact area during loading
of a soda-lime glass sample (from beneath the indenter
tip, looking through the glass sample) at a load of 12.4 N
and Fig. 8(b) is the residual impression of a 12 N inden-
tation on the same sample. Shear faults can be observed
directly outside the contact area (the dark area) in
Fig. 8(a) as bright areas aligned along the contact. The
residual impression in Fig. 8(b) is accompanied by radial
and lateral cracking, which initiated during the early and
late stages of unloading,13 respectively. The length of the
diagonals in Figs. 8(a) and 8(b) are the same a ≈ 33 ␮m.
Radial cracking during loading initiated at 42 N, but the
length of the diagonals, though much more difficult to
measure, remained the same as those measured from the
residual impression. On sapphire, radial cracks initiated
at 5 N during loading, as seen in Fig. 8(c), which shows
the contact under a load of 8 N. The residual impression
shown in Fig. 8(d) is from an 8.3-N indentation and has
approximately the same diagonal length (a ≈ 16 ␮m as
opposed to 15 ␮m under a load of 8 N). The slight dis-
crepancy between the values of the loads used for the
in situ and residual impressions observed for soda-lime
glass and sapphire are due to the displacement-control
nature of the indenter. This discrepancy is rendered in-
consequential by the quantitative analysis below. Finally,
Figs. 8(e) and 8(f) show the contact on NaCl under a load
of 4.3 N and the residual impression of an indentation to
the same load. Again, the length of the diagonal is the
same during loading, as they are following complete un-
load, in this case a ≈ 234 ␮m. Though these materials
cover a wide range of modulus and hardness (Table I),
they all show negligible surface recovery of the diago-
nals during unloading.
The phenomenon of surface recovery for trans-
parent materials was examined quantitatively as well.
Figures 9(a)–9(c) show a comparison of the contact area
measured during loading and post-indentation from
the residual impression diagonals on soda-lime glass,
sapphire, and NaCl, respectively, with increasing
indentation load. The two estimates of area coin-
cide within experimental uncertainty for all three
materials (which span the entire range of E* and H in-
vestigated in this study). Empirical linear fits to the
in situ data [Eq. (2)] are indicated by solid lines.
The onset of radial cracking on loading during indenta-
tion of soda-lime glass and sapphire is indicated by ver-
tical dashed lines. Radial cracking during loading did not
appear to significantly affect the linear A-P trend ob-
served in Figs. 9(a) and 9(b) and predicted by geometric
similarity.
FIG. 7. P-h traces of indentations on (a) soda-lime glass, (b) sap-
phire, and (c) NaCl. The P-h traces have been corrected for column
compliance.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
FIG. 8. Images of the contact region during loading (left) and following complete unload (right). (a) The contact region in soda-lime glass under
12.4 N load. (b) The residual impression of a 12 N indentation in soda-lime glass. (c) The contact region in sapphire under 8 N load (note the radial
cracking). (d) The residual impression of an 8.3 N indentation in sapphire. (e) The contact region in NaCl under 4.3 N load. (f) The residual
impression of a 4.3 N indentation in NaCl.
The area function, Eq. (7), was calibrated to determine
the effective equivalent cone angle ␣eff and effective tip
radius Reff using the procedure outlined in Oliver and
Pharr6 and Thurn and Cook.8 Four materials were used in
the calibration procedure: soda-lime glass, silicon,
Al2O3–TiC, and sapphire (Table I). The area function fits
the data well, following four iterations, over two orders
of magnitude of contact depth, as shown in Fig. 10. The
area function parameters obtained were ␣eff ס 70.40°
and Reff ס 3.30 ␮m. The effective equivalent cone angle
can be compared with the ideal equivalent cone angle of
a Vickers indenter of 70.30°. The area calculated from
the P-h curves and Eqs. (6) and (7) for a variety of ce-
ramic materials (including the calibration materials) is
compared with the corner-to-corner area measured post-
indentation in Fig. 11. The agreement is excellent over
three orders of magnitude of area. The area calculated in
this manner also compares well to the area observed
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
during loading, as shown in Fig. 9. Note that the contact
shown in Fig. 12 as a function of h. (The ␤ values for
h < 2Reff are probably inaccurate; area functions do not
work well in this domain, rendering standard deconvo-
lution methods uncertain.8) Shown on Fig. 12 as horizon-
tal dashed lines are ␤ ס ␲/2 from Love’s elastic solution
for a cone,26 ␤ ס 0.91 from experimental measurements
of elastic recovery of Vickers indentation impressions,23
and ␤ ס 0.77, which is a lower limit on ␤ proposed by
Marx and Balke based on elastic-plastic finite element
simulations.54 Several features of Fig. 12 are striking.
First, ␤ is an increasing function of h for all three
area at peak load is predicted well without using correc-
tion factors to Eq. (3) that would (i) account for the non-
circular projected area shape of the Vickers indenter52
and (ii) account for the horizontal displacements of ma-
terial not considered in Sneddon’s analysis.53
The area function, Eq. (7), was also used to calculate
hc from the area measured during loading (plotted in
Fig. 9 for soda-lime glass, sapphire, and NaCl). These
values of hc were then compared to the measured total
displacement h (minus the effect of column compliance)
to obtain estimates of ␤ (ס h/hc) during loading, as
FIG. 10. Area function calibration plot: ␣eff ס 70.40° and
R
eff ס 3.30 ␮m used in Eq. (7).
FIG. 9. Projected contact area measured in situ, post-indentation
(corner-to-corner), and calculated from the P-h data with increasing
load for (a) soda-lime glass, (b) sapphire, and (c) NaCl. The vertical
dashed line denotes the threshold for radial cracking on loading. Solid
lines are empirical linear fits to the in situ data [Eq. (2)].
FIG. 11. Calculated contact area at peak load compared with the area
measured from the residual indentation impressions on a variety of
ceramic materials.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
materials, although it levels off for h > 20 ␮m in NaCl.
values and are independent of contact depth. The litera-
ture modulus and hardness values used for comparison
were obtained from the references listed in Table I. Hard-
ness and moduli determined using the corner-to-corner
area from the residual indentation impressions following
the indentation event were identical, within experimental
error, to those shown in Figs. 13(a) and 13(b), as may be
confirmed from the area comparison shown in Fig. 11.
The measured modulus and hardness are compared
with the literature values shown in Table I in Figs. 14(a)
and 14(b), respectively. Both the measured modulus and
hardness are lower than the literature values for sapphire
and polycrystalline alumina, with deviations between
15% and 30%. However, this high deviation was not
seen in an equally stiff and hard material, Al2O3–TiC,
which, like most of the materials, deviated 10% or less
from the literature values (using an assumed Poisson’s
Second, ␤ decreases with decreasing E (Table I) but is
not an obvious function of E/H(E/H ס 279 for NaCl,
22.9 for sapphire, and 11.7 for soda-lime glass, all based
on experimental measurements shown in Sec. IV) On
average, ␤ > 1 as expected for soda-lime glass and sap-
phire (elastic sink-in). However, for the extremely duc-
tile NaCl, ␤ < 1, indicating pileup around the indenter
during loading.
C. Modulus and hardness
The plane–strain modulus can be determined from
Eqs. (3) and (4) if the unloading stiffness and projected
area at peak load are known. Similarly, the hardness can
be determined from Eq. (1). The area A can, in this
case, be determined two different ways, as emphasized in
Fig. 11. It can be measured following the indentation
event or determined from Eqs. (6) and (7). The plane–
strain modulus determined using the area function pa-
rameters given above is shown with contact depth for a
variety of ceramics in Fig. 13(a). The average moduli
agree with the accepted values for most materials (see
below) and are nearly independent of contact depth. The
data for barium titanate and soda-lime glass appear to
have slight trends of decreasing E* with increasing hc;
possibly due to the fact that the frame compliance was
determined from the average response of four different
materials. Similarly, the hardness obtained from the area
function is shown in Fig. 13(b). Again, most of the meas-
ured hardness values are in agreement with the literature
FIG. 12. Evolution of ␤(סh/hc) during loading of soda-lime glass,
sapphire, and NaCl. The elastic solution (␤ ס ␲/2),26 a proposed
lower limit (␤ ס 0.77),54 and an experimentally obtained value of
␤(ס 0.91)23 are also shown as horizontal dashed lines. The data for
h < 2Reff (solid vertical line) are probably inaccurate.8
FIG. 13. (a) Plane–strain modulus determined using the area function
with contact depth for a variety of ceramics. (b) Hardness determined
using the area function with contact depth for a variety of ceramics.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
ratio to convert E* to E). In particular, the modulus and
radius had a drastic effect on the loading portion of the
load–displacement trace. Figure 15 displays the loading
and unloading curves for a 13 N indent on soda-lime
glass on logarithmic scales. The abscissa h is h for the
loading data and h − hf for the unloading data (where hf
was determined directly from the P-h data). The solid
line overlapping the loading data is a line of slope two,
indicating perfect quadratic loading. This line begins to
deviate from the data below about h ס 7.5 ␮m, about
twice the effective tip radius. The deviation is high-
lighted in Fig. 15 by a dashed vertical line at h ס 2Reff.
It has been shown that P-h traces in the elastic limit at
very small displacements can be quantified by either de-
scribing the indenter as a sphere of radius R or as a
blunted conical indenter of tip radius.5,55,56 Theoretical
investigations of tip-blunting (or tip-rounding) have
shown that the entire P-h trace is modified by the pres-
ence of a finite tip radius and not just the small-
displacement region of the P-h trace where the indenter
is approximately spherical.33–34,57,58 Cheng and Cheng59
suggested that the loading curve is best described by a
second-order polynomial and used such fits to describe
finite element results for conical indentation. The dashed
line in Fig. 15 is an empirical second-order polynomial
fit to the loading data and describes it well over the entire
range of h. Modifications have been made to the qua-
dratic P-h relationship on loading [Eq. (2)] accounting
for both the elastic–plastic nature of loading and the ef-
fects of tip radius.60
hardness calculated for NaCl were in close agreement
with the literature values despite the effects of pileup
observed during the in situ experiments.
IV. DISCUSSION
The calibration method of Oliver and Pharr, designed
for analysis of nanoindentation P-h data, can readily be
applied to macroscopic-scale indentation analysis. Uti-
lizing the method of Oliver and Pharr for extraction
of mechanical properties has the additional advantage of
identifying the effective properties of the indenter tip via
a two-parameter area function. The tip here was found to
have an effective tip radius of 3.30 ␮m. This finite tip
FIG. 15. P-h curve for a 13 N indentation on soda-lime glass in loga-
rithmic coordinates. h* ס h for the loading curve and h* ס h − hf for
the unloading curve. The solid line of slope two is representative of
perfect quadratic loading while the dashed line is a second-order poly-
nomial fit to the loading curve. The vertical dashed line is at h ס 2Reff
The dotted line is a straight-line fit to the unloading data. (The P-h data
have been corrected for the effects of column compliance.)
.
FIG. 14. Measured (a) modulus and (b) hardness compared with the
literature values. Assumed Poisson’s ratios are as follows: 0.17 for
soda-lime glass, 0.06 for Si,40 and 0.2 for all others.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
A straight-line fit of the unloading data (in the loga-
n ס 1.59 for the raw and corrected data, respectively.
The calculated modulus and hardness values were in-
creased by 28% and 5%, respectively, on correcting the
raw P-h data for column compliance.
rithmic coordinates) in Fig. 15 resulted in a power-law
dependence of n ס 1.37 and is shown as a dotted line.
However, when a nonlinear curve fit was performed us-
ing Eq. (5) on the original load–displacement unloading
data (also corrected for column compliance), a power-
law dependence of n ס 1.53 was obtained, as suggested
by the experimental P-h data of Oliver and Pharr.6 The
straight-line fit to the unloading data on logarithmic axes
was weighted more strongly by the data near h* ס 0,
where identification of A is critical, and the nonlinear fit
was weighted more strongly by the data near the peak
load and displacement—the region of the P-h curve used
to deduce the stiffness. The data of Fig. 15 then sup-
port the idea that a nonlinear power-law fit to experi-
mental data on linear axes is recommended because
it weights the data near the peak load and displacement
and provides a greater degree of flexibility by allowing
hf to vary.
The unloading curve was also more susceptible to
column compliance effects than the loading curve.
Figure 16 is a logarithmic plot of the loading and unload-
ing data for a 31 N indentation on Al2O3–TiC. The solid
data points are the raw data and the open data points have
been corrected for column compliance. The abscissa la-
bel h* is as defined above for Fig. 15. Both the raw and
compliance-corrected loading data appeared quadratic
for h > 2Reff, as in Fig. 15. The raw and corrected un-
loading data, however, had different power dependen-
cies. When a nonlinear curve fit was performed using
Eq. (5) the power dependencies were n ס 1.45 and
V. CONCLUSIONS
DSI at macroscopic loads was performed on a set of
ceramic materials encompassing a wide range of modu-
lus E, hardness H, and E/H values. The custom-built
indenter allowed continuous load–displacement, P-h,
measurements to be made during Vickers indentation at
loads up to 100 N. The macroscopic scale of the inden-
tations, along with the capability of the instrument to be
mounted on top of an inverted optical microscope,
allowed direct verification of the invariance of contact-
diagonal lengths at their peak–load values during inden-
tation unloading, an assumption implicit in post-indentation
estimation of hardness from residual impression dimen-
sions. Optical measurement of the residual impression
dimensions and associated contact areas also enabled a
simple implementation of the Oliver–Pharr6 iteration
scheme for simultaneous calibration of indenter frame
compliance and indenter tip shape. This scheme, based
on Sneddon’s analysis7 of the relationship between
modulus, contact stiffness and projected contact area was
then used to extract E and H estimates directly from the
unloading P-h traces, without direct reference to obser-
vations of impression dimensions, in much the same
manner as “nanoindentation” tests. The estimated values
were found to be in agreement with independent meas-
urements and the importance of adequate specimen
clamping and frame compliance correction, along with
knowledge of the analytically applicable displacement
range set by the tip shape in obtaining this agreement
were all highlighted. In particular, a two-parameter area
function was used here for the tip, providing an estimate
of the effective tip radius (about 3.3 ␮m for the commer-
cial Vickers diamond used); indentation displacements
smaller than about twice the tip radius were observed to
violate the quadratic loading response associated with
geometrically similar contacts.
Applications of macroscopic DSI beyond estimations
of E and H for homogeneous ceramics are vast, and some
were identified here. Quadratic loading was observed at
large displacements and in situ observations revealed that
it was maintained in materials that exhibited radial crack-
ing during indentation contact. Direct observation of the
contact impressions combined with P-h measurements
also allowed the parameter ␤, the total:contact depth ra-
tio, to be measured, revealing a range of ␤ values asso-
ciated with material-dependent indentation plasticity.
Instrumented indenters for DSI at macroscopic loads
are easy to build and use and can thus be customized
easily for specific applications. The only qualifications,
FIG. 16. P-h curve for a 31 N indentation on Al2O3–TiC in logarith-
mic coordinates. The abscissa is as defined in Fig. 15. The open sym-
bols have been corrected for the effects of column compliance, and the
solid symbols have not.
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J. Thurn et al.: Depth-sensing indentation at macroscopic dimensions
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films, and porous materials without worrying about arti-
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ACKNOWLEDGMENTS
The authors wish to thank Yvete Toivola and Matthew
Cunningham for helpful discussions. Supplemental fi-
nancial support for Jeremy Thurn was provided by the
Coating Process Fundamentals Program through the Uni-
versity of Minnesota Industrial Partnership for Research
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