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Herbert2001.pdf
Thin Solid Films 398 – 399 (2001) 331–335
On the measurement of stress–strain curves by spherical indentation
E.G. Herberta,b,*, G.M. Pharrc, W.C. Oliverd, B.N. Lucase, J.L. Hayd
a
MTS Nano Instruments Innovation Center, Oak Ridge, TN, USA
b
The University of Tennessee, Knoxville, TN, USA
c
University of Tennessee and ORNL, Knoxville, TN, USA
d
MTS Nano Instruments Innovation Center, Oak Ridge, TN, USA
e
Fast Forward Devices, Knoxville, TN, USA
Abstract
It has been proposed that with the appropriate models, instrumented indentation test (IIT) data can be reduced to yield the
uniaxial stress–strain behavior of the test material. However, very little work has been done to directly compare the results from
uniaxial tension and spherical indentation experiments. In this work, indentation and uniaxial tension experiments have been
performed on the aluminum alloy 6061-T6. The purpose of these experiments was to specifically explore the accuracy with which
the analytical models can be applied to IIT data to predict the uniaxial stress–strain behavior of the aluminum alloy. Despite not
being able to reproduce the physical shape of the uniaxial stress–strain curve, the results do indicate that spherical indentation
can be successfully used to establish an engineering estimate of the elastic modulus and yield strength of 6061-T6. 䊚 2001
Elsevier Science B.V. All rights reserved.
Keywords: Nanoindentation; Spherical; Indentation; Stress and strain measurements
1. Introduction
The primary objective of this work was to compare
the stress–strain curve of the aluminum alloy 6061-T6
as determined by spherical indentation and uniaxial
tension. To accomplish that goal, existing indentation
models were used to develop what we felt to be the
most appropriate indentation test method. Developing
the most appropriate load–time algorithm and the most
accurate and meaningful way to establish the point of
contact and the instrument load frame stiffness were
among the most critical aspects of building the test
method. Each of these issues plays a critical role in
obtaining the most precise and accurate mechanical
property measurements possible.
Developing the test method was an iterative process.
The final verification of the models and our test method
* Corresponding author. Tel.: q1-865-481-8452; fax: q1-865-4818455.
E-mail addresses: [email protected] (B.N. Lucas),
[email protected] (E.G. Herbert), [email protected] (G.M. Pharr),
[email protected] (W.C. Oliver).
was based on their ability to determine the elastic
modulus of the standard reference material, which in
this case was fused silica. Young’s modulus was chosen
to benchmark the models and test method performance
because it is an intrinsic material property and, as
indicated by Eqs. (5)–(9), it is a direct verification of
whether or not the models and test method are calculating the correct contact area.
Once the models and test method demonstrated the
ability to correctly measure the elastic modulus of the
standard reference material, they were applied to the
aluminum alloy 6061-T6. The primary questions were,
can the models and test method accurately and precisely
determine Young’s modulus, E, and the yield strength,
s y?
2. Theory
The models chosen to reduce the indentation data
were that of Hertz w1x, Oliver and Pharr w2x and Tabor
w3x. Hertz’s analysis, despite being based on the assumption of paraboloids in elastic contact, was chosen over
0040-6090/01/$ - see front matter 䊚 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 4 3 9 - 0
E.G. Herbert et al. / Thin Solid Films 398 – 399 (2001) 331–335
332
Sneddon’s w4x specific solution of spheres in elastic
contact for two reasons: in the limit of small displacements (2hcR4h2c ), Hertz’s solution is very similar to
Sneddon’s and Hertz’s solution is algebraically much
more simple. Hertz’s analysis provides three expressions
that are used in this investigation. First is the simple
relationship between the total displacement into the
sample, h, and the contact depth, hc,
(1)
hs2hc
This is one expression among three that will later be
used to indicate the yield point. Second is the following
relationship between the load, P, and the contact radius,
a,
Ps
4Era3
3R
(2)
where Er is the reduced or indentation modulus and R
is the nominal radius of the indenter tip. Third is Hertz’s
relationship between the total displacement into the
sample and contact radius,
2
(3)
hsa yR
Combining Eqs. (1)–(3), the relationship between
load and contact depth can be simplified to
Ps
8y2
3
EryRh
3y2
c
(4)
The importance of this expression is twofold. First, it
provides a meaningful way to establish the point of
contact and secondly it can be used to indicate when
yielding has occurred.
Encompassed within the Oliver–Pharr model is the
work of Pharr et al. w5x, who showed that for all
axisymmetric indenters, there is a constant relationship
between the elastic contact stiffness, S, the projected
area of contact, A, and the reduced or indentation
modulus. The relationship is,
E rs
yp s
2 yA
(5)
In this context, the area is simply computed from the
radius of the contact,
Aspa2
(6)
which is calculated from the following expression,
asy2hcRyh2c
(7)
The contact depth is established from the Oliver–
Pharr model,
hcshy´PyS
(8)
where h is the total measured displacement into the
sample, and ´ is a geometric constant based on indenter
geometry; ´ is 0.75 for a sphere. Furthermore, Er is
related to the elastic modulus of the sample, Es, through
the elastic modulus of the indenter, Ei, and Poisson’s
ratio of the indenter, ni, and the sample, ns:
w 1yni
E rsx
y
Ei
q
1yns zy1
|
Es ~
(9)
Therefore, given that we directly measure the contact
stiffness, the only way to correctly calculate the known
Young’s modulus, E or Es, is to calculate the correct
contact area. It is important to note the contact area is
determined only by the nominal radius of the indenter
tip and the contact depth. There is no empirically derived
‘area function’ associated with the contact area
calculation.
Three important observations of Tabor’s w3x are also
used in this analysis. First is the observation that yielding
occurs when the mean pressure divided by the yield
strength is approximately equal to 1.07,
(10)
pmysyf1.07
The next two observations of Tabor’s go hand-inhand. The true-stress and true-strain as determined from
uniaxial tension data are equivalent to indentation stress
and strain through the following two expressions:
sindentationfpmy3fsuniaxial
tension
´indentationf0.2ayRf´uniaxial
tension
(11)
(12)
It is important to note that Eqs. (11) and (12) only
hold true in the limit of a fully developed plastic contact.
Based on the recent finite element investigations of
Mesarovic and Fleck w6x, this regime occurs when ay
Rf0.16, independent of the magnitude of the nondimensional parameters sy yEr and Poisson’s ratio.
3. Experimental methods
The indentation experiments were performed using a
Nano Indenter䉸 XP with MTS’ continuous stiffness
measurement (CSM) technique. With this technique,
each indent gives the load, displacement, and contact
stiffness as a continuous function of the indenter’s
displacement into the sample w2x. Loading was controlled such that the loading rate divided by the load
was held constant at 0.05ys. Experiments were terminated at the maximum load of the standard XP head,
approximately 70 g. The four spherical tips were singlecrystal sapphire with diameters of 120, 200, 260 and
300 mm. The tensile experiments were conducted on an
MTS 10yGL load frame with a 10 000-lb load cell. The
dog bone samples were pulled at a rate of 0.05 mmys.
4. Results and discussion
Fig. 1 represents the elastic modulus as a continuous
function of the indenter’s displacement into the sample
E.G. Herbert et al. / Thin Solid Films 398 – 399 (2001) 331–335
333
Fig. 1. Fused silica, modulus of elasticity as a function of the indenter’s displacement into the sample.
for the standard reference material, fused silica. The
literature value for fused silica is 72 GPa. Clearly, the
models and method are doing a reasonably good job of
evaluating the correct contact area using just the contact
depth and the nominal radius of the tip. It is worth
noting that all of the data in Fig. 1 are representative of
an elastic contact, indicating that the Oliver–Pharr model
simply reduces to Hertz’s solution in the limit of an
elastic contact. Furthermore, the silica data conformed
very well to Eq. (1); the contact depth is indeed equal
to one-half the total measured displacement into the
sample.
Based on the ability of models and the test method
to successfully determine the elastic modulus of the
standard reference material, the next step was to apply
them to the 6061-T6. Fig. 2 represents the elastic
modulus as a function of the indenter’s displacement
into the sample for the 6061-T6. The average modulus
value obtained from the tensile experiments was
70.09"0.6 GPa. The literature value is 70.1 GPa. With
modulus values obtained from indentation ranging from
67 to 75 GPa, the models and test method appear to be
doing a reasonably good job of determining the elastic
modulus. Furthermore, it is worth pointing out that the
data in Fig. 2 are representative of elastic, elastic–plastic
and fully developed plasticity, thereby providing solid
testimony that the Oliver–Pharr model works well in all
three regimes.
The next parameter of interest is the yield strength.
Which promptly brings about the question, ‘exactly how
can we estimate when the sample has yielded?’ Three
different methods were explored in this work, all of
Fig. 2. 6061-T6, modulus of elasticity as a function of the indenter’s displacement into the sample.
E.G. Herbert et al. / Thin Solid Films 398 – 399 (2001) 331–335
334
Fig. 3. 6061-T6, hcyh as a function of the indenter’s displacement into the sample.
which rely on a deviation from the expected elastic
behavior predicted by Hertz. The three expressions used
were as follows:
1. hc yhs0.5;
2. in the limit of small displacements (2hcR4h2c ),
P2y3
; and
Ž8y2y3.EryR 2y3hc
3. loading slopeyunloading slopes1.
Ž
.
All three relationships are known to hold true in the
limit of an elastic contact. The yield point should be
identifiable by any deviation from any one or all three
expressions. Unfortunately, all three were equally poor
performers in terms of providing any conclusive evidence of the yield point. In comparing the three, the
ratio of contact depth to total depth arguably made the
clearest distinction. Fig. 3 represents that ratio as a
function of the indenter’s displacement into the sample.
Regrettably, the data do not provide any conclusive
evidence of the yield point. As indicated by Hertz’s
analysis, the data for all four spheres should be constant
at 0.5 until the material yields. As illustrated by Fig. 3,
the data somewhat resembles that description but clearly
at shallow depths it is not meeting expectations. Sorting
out exactly what is causing these unexpected results will
be an integral part of future work. Nevertheless, in an
effort to make due with the acquired data, we chose to
pick what we thought represented an upper and lower
limit on the displacement at which yielding had
occurred. In looking at the data from 400 nm towards
the origin, for all four spheres, there is a distinctive
curved shape that terminates at a small plateau or
inflection point. With some confidence, the upper displacement limit for yielding was chosen to be the
second-to-last data point that could be considered as
part of that curved shape. For the 300-mm sphere, the
upper limit is illustrated by the arrow in Fig. 3. Given
justifiable concern as to whether or not the contact was
ever solely elastic, choosing the lower displacement
limit with any measure of confidence was deemed
virtually impossible. Lacking a better alternative, the
lower displacement limit for yielding was simply chosen
to be the data point immediately left of the upper
displacement limit. From the upper and lower limits, a
yield strength range was calculated using Tabor’s w3x
observation that at the point of yielding, the yield
strength is approximately equal to the mean pressure
divided by 1.07 (Eq. (10)),
Pypa2
fsy
1.07
(13)
Plugging in the corresponding upper and lower limits
for a led to the following average estimates of the yield
strength:
● Lower limit: 399"44.5 MPa.
● Upper limit: 426"36.4 MPa.
The average result from the uniaxial tension experiments was 274"2.3 MPa. The literature value is 276
MPa. The lower and upper estimates of the yield strength
from the indentation data are 45 and 54% too high,
respectively. Based on the ambiguity of method used to
determine the yield point, these results are not too
surprising.
Interestingly enough, however, the experimental
results are very well corroborated by the recent finite
element investigations of Mesarovic and Fleck w6x. Their
results indicate that while yielding does occur at the
point pm ysyf1.07, the first evidence of yielding does
not actually occur until pm ysyf1.6. Using their 1.6 as
E.G. Herbert et al. / Thin Solid Films 398 – 399 (2001) 331–335
335
Fig. 4. The true-stress true-strain behavior of 6061-T6 as determined by uniaxial tension and spherical indentation.
opposed to 1.07 brings the lower and upper estimates
of the yield strength to within y3.1 and 3.3% of the
literature value, respectively.
Fig. 4 presents the true-stress true-strain curve for the
aluminum alloy 6061-T6 as determined by uniaxial
tension and spherical indentation. The vertical line
through the plot indicates the point at which Tabor’s
observations, Eqs. (11) and (12), hold true. The line’s
position is representative of the point at which ayRs
0.16. Although the indentation data to the left of the
line does resemble the shape of a stress–strain curve,
current indentation models in the literature do not
provide any relationship between stress and strain in
this regime.
5. Conclusions
With the correct models and careful attention to
experimental detail, spherical indentation can be successfully used to determine the elastic modulus of 6061T6.
With sphere diameters of 120, 200, 260 and 300 mm,
a good engineering estimate of the yield strength of
6061-T6 can be made.
Applying Hertz’s elastic models to spherical indentation data can be successfully accomplished in the limit
of small displacements and truly spherical tip geometry.
With respect to 6061-T6 and fused silica, the Oliver–
Pharr model has successfully demonstrated its ability to
accurately predict the contact depth in all three indentation regimes: elastic, elastic–plastic, and fully developed plasticity.
References
w1x H. Hertz, in: H. Hertz (Ed.), Miscellaneous Papers, Jones and
Schott, Macmillan, London, 1863.
w2x W.C. Oliver, G.M. Pharr, J. Mater. Res. 7 (6) (1992) 1564–
1583.
w3x D. Tabor, Hardness of Metals, Clarendon Press, Oxford, 1951.
w4x I.N. Sneddon, Int. J. Eng. Sci. 3 (1965) 47.
w5x G.M. Pharr, W.C. Oliver, F.R. Brotzen, J. Mater. Res. 7 (1992)
613.
w6x S.D. Mesarovic, N.A. Fleck, Spherical indentation of elastic–
plastic solids, Proc. Royal Soc. Lond. 455 (1999) 2707–2728.
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