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Hay2011.pdf
Strain-Rate Sensitivity (SRS) of Nickel
by Instrumented Indentation
Application Note
Jennifer Hay, Agilent Technologies, Nano-Scale Sciences Division
Verena Maier, Dr. Karsten Durst, and Dr. Mathias Göken,
University of Erlangen-Nuremberg, Department of Material Science and Engineering
Introduction
In many materials, the plastic stress
that can be sustained depends on
strain rate through a power-law
relationship: higher stresses are
sustained with higher strain rates
and vice versa. In a uniaxial tensile
configuration, this relationship
between plastic stress, , and strain
rate, •u, is expressed as
m,
= B*•u
Eq. 1
where B* is a constant and m is the
strain-rate sensitivity (SRS), which
is always greater than or equal to
zero. For materials which manifest
negligible strain-rate sensitivity, m
is near zero, making a constant.
(Sapphire is an example of such a
material.) Materials with greater
strain-rate sensitivity have greater
values of m.
Provided that hardness (H) is
directly related to plastic stress, then
hardness also manifests this same
phenomenon, giving the relation
H = B• m .
Eq. 2
In Eq. 2, B is a constant (though
different in value from B* in Eq. 1)
and • is the indentation strain rate,
1.
defined as the loading rate divided
•
by the load (P /P)1. The strain-rate
sensitivity, m, has the same meaning
and value in Eq. 2 as it does in Eq. 1.
Taking the logarithm of both sides of
Eq. 2 and simplifying yields
In(H) = m • In(• ) + In(B).
Eq. 3
Thus, for many materials, there
is a linear relationship between
the logarithm of hardness and
the logarithm of strain rate, with
the slope being the strain-rate
sensitivity, m.
Lucas and Oliver showed that the
strain-rate sensitivity, m, could be
evaluated by performing a series of
indentations, with each indentation
performed using a different strain
rate [1]. However, the approach of
Lucas and Oliver is problematic,
because indentations at small strain
rates take so long that the results
can easily be dominated by thermal
drift. Recently, Maier et al. showed
that all strain rates of interest may be
executed within a single indentation
test by switching strain rates as the
indenter continues to move into the
material [2].
Strictly, the term ‘indentation strain rate’ refers to the displacement rate divided by the
•
displacement (h /h). However, beginning with the definition of hardness, it is easily shown that
•
•
h /h ≈ 0.5(P/P). Eq. 2 holds true for either definition of strain rate, because the constant (0.5)
difference between the two definitions is simply absorbed into the constant B. Because the Agilent
•
•
G200 NanoIndenter is a force-controlled instrument, it is logistically easier to control P/P than h /h.
•
Thus, in this work, the term ‘strain rate’ refers to P/P, unless specifically stated otherwise.
The protocol proposed by Maier et al.
has a number of practical advantages.
First, the testing time and thermal
drift are minimized by using fast
strain rates when the applied force is
small and slow strain rates when the
applied force is large. To understand
this benefi t, it is important to
understand how a controlled-strainrate experiment works. The forceapplication rate required to maintain a
given strain rate changes with applied
force. For example, let us compare
the force-application rate required
to achieve a strain rate of 0.01/sec
at 1mN and 100mN. Knowing the
•
definition of strain rate (• = P /P),
we calculate the necessary force•
application rate for each situation (P)
as the product of the desired strain
rate and the applied force. When the
applied force is 1mN, we have
•
P = 0.01/sec*1mN = 0.01mN/sec.
When the applied force is 100mN,
we have
•
P = 0.01/sec*100mN = 1mN/sec,
which is much faster. Though the
same strain rate is achieved in
both cases, the associated force
rate is much higher in the second
case, because the applied force is
much higher. Thus, it may take a
prohibitively long time to examine a
small strain rate when the applied
force is small, but that same small
strain rate can be examined rather
quickly when the applied force is
large. The protocol suggested by
Maier et al. takes advantage of this
reality by examining the largest strain
rate at the beginning of the test
(when the applied force is small) and
by examining progressively smaller
strain rates as the applied force
increases. In this way, both testing
time and thermal drift are minimized.
2
The protocol of Maier et al. has been
implemented in a new NanoSuite test
method; this application note reports
the results obtained with this new
method on a nickel standard reference
material (SRM) produced by the U.S.
National Institute of Standards and
Technology (NIST).
The protocol of Maier et al. has a
second important benefi t: because all
strain rates of interest are examined
in every test, it is possible to map out
the spatial distribution of strain-rate
sensitivity. Although this capacity
will not be demonstrated in this note,
Maier et al. measured local strain-rate
sensitivity in and around a bond layer
in roll-bonded aluminum.
Sample
The sample tested in this work was
a NIST standard reference material
(SRM) for Vickers hardness. The
sample consists of a 1.35 cm square
test block of electrodeposited bright
nickel, approximately 750 microns
thick, on an AISI 1010 steel substrate,
mounted and highly polished in a
thermosetting epoxy. A template
certificate for this kind of sample can
be found on the NIST website [3]. The
qualities which make this sample ideal
as a Vickers SRM also make it ideal
for the present demonstration. It has a
smooth surface, is resistant to tarnish
and corrosion, and has a small grain
size. These qualities are important,
because ideally, changes in strain
rate should be the only explanation
for the observed changes in hardness.
Changes in hardness due to other
factors such as surface layers,
indentation size effect, and constraint
influence can all compromise the
validity of the measured strain-rate
sensitivity.
Equipment
Parameter
An Agilent G200 NanoIndenter with
a Berkovich indenter was used for
all testing. The Continuous Stiffness
Measurement Option (CSM) was also
used in order to achieve hardness
and elastic modulus as a continuous
function of penetration depth [4].
Surface Approach Velocity
Surface Approach Distance
Value
Units
25
nm/s
1000
nm
5
nm
Frequency Target
45
Hz
Poissons Ratio
0.3
Harmonic Displacement Target
Displacement, Initial
Strain Rates, How Many?
1100
nm
3
Test Method
Displacement per Rate
150
nm
Twelve indentation tests were
performed using the test method
“G-Series XP CSM Strain-Rate
Sensitivity.” Table 1 summarizes
testing inputs. This test method
allows the user to prescribe a
penetration that must be achieved
prior to strain-rate cycling
(Displacement, Initial). This initial
penetration is used to achieve a
penetration depth that is large enough
so that no further changes in hardness
are expected due to indentation size
effect, surface inhomogeneities,
tip anomalies, etc. Once this initial
penetration has been achieved, the
method prescribes cycling between a
test strain rate and a base strain rate.
The test strain rate is executed in the
first part of the cycle, and the base
strain rate is executed in the second
part of the cycle. The return to the
base strain rate after each test strain
rate provides a means for confirming
that hardness is not changing with
increasing penetration for any reason
other than the changing influence of
strain rate. Figure 1 shows the strainrate history for each indentation test
on the Ni SRM.
Strain Rate, Maximum
0.05
1/s
Strain Rate, Minimum
0.001
1/s
Table 1. Summary of inputs used to measure the strain-rate sensitivity of a Ni
Vickers SRM by the method “G-Series XP CSM Strain-Rate Sensitivity.”
Figure 1. Strain-rate cycling imposed on the test sample. In the first part of the cycle, the test strain
rate is imposed. In the second part of the cycle, the base strain rate is imposed.
3
Results and Discussion
Test
1
2
3
4
5
6
7
8
9
10
11
12
Avg.
Std. Dev.
Table 2 summarizes the most
important results. The elastic
modulus (E) of the Ni SRM was
measured to be 229±3 GPa, and
the strain-rate sensitivity (m) was
measured to be 0.021±0.002. Table 3
is a survey of strain-rate-sensitivity
values measured by others for finegrained Ni.
Figure 2 shows the continuous
elastic modulus during strainrate cycling for one typical test.
As expected, the modulus did not
change significantly during strainrate cycling. Modulus is reported
for each cycle by averaging the
continuous measurements which fall
within 80-90% of the displacement
range for the base-strain-rate
segment of the cycle. In Figure 2,
these measurements are plotted
as green data points. The modulus
value reported for each test in
Table 2 is the average of the three
cycle-level results for that test. The
average over all tests, 229 GPa, is
15% higher than the nominal value of
200 GPa for pure nickel. One possible
explanation for the discrepancy is
that the Oliver-Pharr model for the
contact area may overestimate
the true contact area. Because the
calculation of elastic modulus by
indentation goes as the inverse of
the square root of the contact area,
an underestimation of the contact
area leads to an overestimation
of the modulus. Finite-element
simulations of indentations into a
material with nickel-like properties
could be used to further investigate
this explanation, because finiteelement simulations allow a
comparison between contact area
determined by the Oliver-Pharr model
and contact are determined from the
finite-element mesh.
E, GPa
226.8
225.5
228.8
230.5
228.3
234.5
228.4
235.5
229.9
224.6
229.8
228.9
229.3
3.2
m
0.0207
0.0207
0.0238
0.0207
0.0228
0.0212
0.0187
0.0228
0.0220
0.0179
0.0189
0.0210
0.0209
0.0018
Table 2. Summary of Results.
Source
This work
Maier et al. [2]
Maier et al. [2]
Shen et al. [5]
Dalla Torre et al.
[6, 7]
Wang et al. [8]
Sample
Ni Vickers SRM
Nanocrystalline Ni
Nanocrystalline Ni
Nanocrystalline Ni
Nanocrystalline Ni
Method
m
Indentation
0.021
Indentation
0.019
Uniaxial creep (compression)
0.016
Uniaxial creep (tension)
0.016–0.045
Uniaxial creep (tension)
0.010–0.030
Nanocrystalline Ni Uniaxial creep (tension)
0.019
Table 3. Survey of SRS (m) values measured by others on fine-grained Ni.
Figure 2. Modulus during strain-rate cycling for one typical test. As expected, modulus is not
sensitive to strain rate.
4
Figure 3. Hardness during strain-rate cycling. Sensitivity to strain rate is evident. Black symbols
denote data used to calculate the hardness for each test strain rate. Green symbols denote data
used to calculate the hardness for each base strain rate.
Figure 4. Plot of ln(H) vs. ln(• ) for each of twelve tests. Slope of ln(H) with respect to ln(• ) for
each test gives the strain-rate sensitivity, m, for that test. Linearity of these data demonstrates
that strain-rate sensitivity for this material is well modeled by Eq. 2.
5
Figure 3 shows the continuous
hardness measured during strain-rate
cycling for a typical test. (These data
are from the same test for which the
modulus is plotted in Figure 2.) The
influence of changing strain rate is
obvious. At each change, there is a
transient in the hardness response
as the microstructure adjusts to the
new rate. For each test strain rate,
the hardness values within 80%-90%
of the displacement range for the
segment are averaged to report a
single value of hardness; data within
this range for each cycle are plotted
as black symbols in Figure 3. Figure 4
shows Ln(H) vs. Ln(° ) for all 12
tests. The linearity of these results
supports the hypothesis that this
material is well described by Eq. 2. For
each test, the strain-rate sensitivity,
m, is calculated as the slope of
Ln(H) vs. Ln(•) and reported in
Table 2. The value for strain-rate
sensitivity obtained by averaging over
all 12 tests (m = 0.021±0.002) is well
within the range of SRS values that
have been measured by others for
fine-grained Ni (Table 3).
A hardness value for each
implementation of the base strain
rate was also determined. For each
base strain rate, the hardness values
within 80%-90% of the displacement
range for the segment are averaged to
report a single value of hardness for
that segment; in Figure 3, the included
hardness values are plotted as green
symbols. For all 12 tests, Figure 5
shows hardness associated with the
base strain rate for each cycle. The
lack of any trend in hardness with
cycle number confirms that hardness
is not changing with depth when the
same strain rate is applied.
Conclusions
An experimentally robust test method
has been implemented in NanoSuite
for measuring strain-rate sensitivity
by instrumented indentation. The
method overcomes problems
associated with long testing times by
imposing small strain rates only when
the applied force is large. Using this
method, the strain-rate sensitivity
of a sample of nickel sold by NIST
as a Vickers SRM was measured to
be m = 0.021. This value is in good
agreement with values obtained by
others on similar materials using both
instrumented indentation and uniaxial
creep testing.
Figure 5. Hardness measured during the base-strain-rate segment of each cycle. Lack of any trend
with cycle number demonstrates that the same hardness is measured when the same strain rate is
applied, despite increasing penetration.
6
References
1. Lucas, B.N. and Oliver, W.C.,
“Indentation Power-Law Creep of
High-Purity Indium,” Metallurgical
and Materials Transactions A-Physical
Metallurgy and Materials Science 30(3),
601-610, 1999.
2. Maier, V., Durst, K., Mueller, J.,
Backes, B., Hoppel, H., and Göken, M.,
“Nanoindentation strain rate jump
tests for determining the local strain
rate sensitivity in nanocrystalline Ni
and ultrafine-grained Al,” Journal of
Materials Research 26(11), 1421-1430,
2011.
3. NIST Certificate Standard Reference
Material 1896a Vickers Microhardness
of Nickel. [cited 2011 October
10, 2011]; Available from: http://
ts.nist.gov/MeasurementServices/
ReferenceMaterials/upload/1896a.pdf.
4. Hay, J.L., Agee, P., and Herbert, E.G.,
“Continuous Stiffness Measurement
during Instrumented Indentation
Testing,” Experimental Techniques
34(3), 86-94, 2010.
5. Shen, X., Lian, J.S., Jiang, Z., and Jiang,
Q., “High Strength and High Ductility
of Electrodeposited Nanocrystalline
Ni with Broad Grain Size Distribution,”
Material Science and Engineering A
487, 410, 2008.
6. Dalla Torre, F., Van Swygenhoven, H.,
and Victoria, M., “Nanocrystalline
Electrodeposited Ni: Microstructure
and Tensile Properties,” Acta Materialia
50, 3957, 2002.
7. Dalla Torre, F., Spätig, P., Schäublin,
R., and Victoria, M., “Deformation
Behavior and Microstructure of
Nanocrystalline Electrodeposited and
High Pressure Torsioned Nickel,” Acta
Materialia 53, 2337, 2005.
8. Wang, Y.M., Hamza, A.V., and Ma, E.,
“Temperature-Dependent Strain-Rate
Sensitivity and Activation Volume in
Nanocrystalline Ni,” Acta Materialia
54, 2715, 2006.
7
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