389
Nanoindentat
16. Nanoindentation: Localized Probes
of Mechanical Behavior of Materials
David F. Bahr, Dylan J. Morris
16.1 Hardness Testing:
Macroscopic Beginnings ........................
16.1.1 Spherical Impression Tests:
Brinell and Meyers .......................
16.1.2 Measurements of Depth
to Extract Rockwell Hardness .........
16.1.3 Pyramidal Geometries
for Smaller Scales: Vickers Hardness
16.2 Extraction of Basic Materials Properties
from Instrumented Indentation .............
16.2.1 General Behavior
of Depth Sensing Indentation ........
16.2.2 Area Functions .............................
16.2.3 Assessment of Properties
During the Entire Loading Sequence
389
390
390
391
392
392
394
395
16.3 Plastic Deformation at Indentations ....... 396
16.3.1 The Spherical Cavity Model ............ 397
16.3.2 Analysis of Slip Around
Indentations ............................... 398
16.4 Measurement of Fracture
Using Indentation ................................ 399
16.4.1 Fracture Around Vickers
Impressions................................. 399
16.4.2 Fracture Observations
During Instrumented Indentation .. 400
16.5 Probing Small Volumes to Determine
Fundamental Deformation Mechanisms.. 402
16.6 Summary ............................................. 404
References .................................................. 404
16.1 Hardness Testing: Macroscopic Beginnings
Out of the many ways in which to measure the mechanical properties of a material, one of the most common
is the hardness test. While more basic materials properties can be determined from testing materials in uniaxial
tension and compression (along with shear testing), the
hardness test remains useful in both industrial and laboratory settings for several reasons. First, the testing
method is very quick and easy, and allows many measurements on a given sample. Secondly, the equipment
for the test methods and required sample preparation
Part B 16
This chapter focuses on mechanical probes of
small volumes of materials using contact based
testing methods performed on instruments designed to measure mechanical properties, rather
than those which are inherently developed for
scanning probe microscopy. Section 16.1 consists
of basic information used in the measurement
of localized mechanical properties and provides
a brief review of engineering hardness testing,
followed in Sect. 16.2 by a discussion of the basis
for instrumented indentation testing to determine
elastic and plastic properties of the material being tested: the common modulus and hardness
techniques. More advanced methods for using localized probes beyond determining basic elastic
and plastic properties, including developments in
the analysis of the deformation zone around indentations and the resulting information that can
be garnered from this information, are covered
in Sect. 16.3. Methodologies that are used to estimate other mechanical properties such as fracture,
are covered in Sect. 16.4, which focuses on bringing instrumented indentation testing into realms
similar to current macroscopic mechanical testing. Finally, the chapter concludes in Sect. 16.5
with methods that use nanoscale indentation
testing to determine more fundamental properties, such as the onset of dislocation activity,
size-dependent mechanisms of deformation, and
activation volumes.
390
Part B
Contact Methods
Part B 16.1
is relatively inexpensive, making the tests attractive for
small laboratories with limited budgets. Finally, and
most importantly, the tests can be made on the actual
parts and is sensitive to lateral position. For large parts,
a small indentation is usually nondestructive, and allows for quality control checks of each part. Sectioning
the sample allows testing of parts which have varying properties through the thickness of the part, such
as case hardened gears. While hardness may not be
as pure a measurement of a material’s properties as
uniaxial testing, the advantages of these methods ensure that the test methods will continue to be used in
both industrial and research settings for many applications.
Traditional hardness tests have been used for
decades to measure the resistance of a material to
deformation. In the late 18th and early 19th century
several attempts were made to rank the deformation
of materials by scratching one material with another.
If the scratching material marked the tested material,
the scratching material was considered harder than the
scratched material. While this was useful for comparative purposes (and is still referred to in the Mohs
hardness scale), there remained the problem of quantifying hardness. In the late 19th century work by
Hertz [16.1] and Auerbach [16.2] brought about early
examples of static indentation testing. In these cases,
the samples (either balls or flat plates) were probed with
a ball of a given material, and the deformation was
measured by considering the contact area between the
ball and sample.
16.1.1 Spherical Impression Tests:
Brinell and Meyers
It was not until the early 1900s that Brinell [16.3] presented a standard method of evaluating hardness, based
on applying a fixed load to a hard spherical indenter tip
into a flat plate. After 15–30 s, the load was removed
and the diameter of the impression was measured using optical microscopy. The Brinell hardness number
(BHN) is defined as
BHN =
π D2
2P
,
d 2
1− 1− D
(16.1)
where P is the applied load, D the diameter of the indenter ball, and d is the chordal diameter of the residual
impression. Note that this method evaluates the load
applied to the surface area of the residual impression.
However, the Brinell test has been shown to be affected
by both the applied load and diameter of the ball used
for the indenter.
The idea that hardness should be a materials property, and not dependent on the test method, led to the
observations that the parameter which remains constant
for large indentations appears to be the mean pressure,
defined as the load divided by the projected contact area
of the surface. Meyer suggested in 1908 [16.4] that the
hardness should therefore be defined as
4P
,
(16.2)
H=
πd 2
where H is the hardness. For a confirmation of the work
of Meyer (and a report of the ideas proposed in English) the reader is referred to the study of Hoyt [16.5] in
1924. This type of hardness measurement is alternately
referred to as the mean pressure of an indentation,
and denoted by either p0 or H in the literature. Tabor
describes in detail one of the direct benefits of utilizing spherical indentation for determining the hardness
of a material[16.6]. As the indentation of a spherical
indentation progresses, the angle of contact of the indentation changes. From extensive experiments in the
1920s to 1950s, it was found that the effective strain of
the indentation εi imposed by a spherical tip could be
approximated by
d
(16.3)
.
D
Similarly, for materials such as copper and steel, H is
approximately 2.8 times the yield stress. This allowed
the creation of indentation stress–strain curves; by applying different loads to spherical tips and measuring
the impression diameters one found that the hardness of
a material which work hardens increased with increasing indentation area for spherical tips.
εi ≈ 0.2
16.1.2 Measurements of Depth
to Extract Rockwell Hardness
Both the Brinell and Meyer hardness methods require
each indentation to be observed with an optical microscope to measure the diameter of the indentation. This
makes the testing slow and requires a skilled operator
to carry out the test, with the result that the method is
more appropriate to a laboratory setting than industrial
settings. The Rockwell hardness test, designed originally by Stanley P. Rockwell, focused on measuring
the depth an indenter penetrated into a sample under
load. As the depth would correspond to an area of
contact if the geometry of the indenter was know, it
was thought that measuring the depth would provide an
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
accurate and automatic measurement of hardness. For
a detailed description of the history of the development
of hardness testing, the reader is referred to the work of
Lysaght [16.7].
In standardizing the Rockwell test methods, it became common to use a conical diamond indenter tip
which was ground to an included angle of 120◦ when
testing very hard materials. This indenter tip, referred
to as a Brale indenter, is the basis for the common
Rockwell C, A, and D scale tests. During a Rockwell
test, the sample is loaded to a preload of 10 kg to alleviate effects of surface roughness. After this preload
is applied, a larger load is applied, referred to as the
major load. The depth measured during the test is the
change in distance between the penetration of the tip
at preload and after the application of the major load.
Effectively, this is the first time that depth sensing was
instrumented during indentation testing, and eventually
leads to the development of instrumented indentation
methods.
16.1.3 Pyramidal Geometries
for Smaller Scales: Vickers Hardness
where DPH is the diamond pyramid hardness number,
P is the load applied in grams, and now d is the length
of the diagonals in mm (usually the average of the two
diagonals). This is analogous to the Brinell test, in that
the DPH is the ratio of the load to the surface area of
the indentation. The main advantages of this method is
that there is a continuous scale between very soft and
very hard materials, and that the DPH is constant over
a wide range of loads until very low loads (less than
50–100 g) are reached. However, the test still required
each indentation to be measured optically to determine
the hardness. The DPH can be converted to a mean
pressure by simply using (16.2), by substituting the projected area for the ratio.
The DPH tests were then adapted to loads of less
than 1000 g, which are now commonly referred to as microindentation techniques. This allowed testing of very
small areas around the indentation. In addition, as the
tests began to gain use in probing cross sections of case
hardened materials, the lateral resolution was altered by
adding the Knoop indenter tip, which is a four-sided
diamond with an aspect ratio of 7:1 between the diagonals [16.6]. The small areas deformed by the DPH
and Knoop methods allowed testing of thin coatings and
of different microstructural features, such as areas of
ferrite and martensite in a steel alloy.
Other indenter geometries have been developed, in
particularly the Berkovich tip, which is a three-sided
pyramid with the same projected contact area as the
Vickers tip [16.9]. This results in a face angle between an edge and the opposing face of approximately
65.3◦ . The driving force towards using the Berkovich
geometry over conical or Vickers pyramids is that fabricating diamond tips with extremely small root radii
is challenging. As three planes intersect at a point,
experimentally it is easier to fabricate the Berkovich geometry. One should be aware that even the most finely
fabricated diamond tips do not reach the pyramidal geometries at their apex, and commonly are approximated
by an effective tip radius [16.10]. Experimental measurements of commercially available tips [16.11] show
that the tip radius of Berkovich indenters is commonly
less than 200 nm. If a simple geometrical model of an
axisymmetric indenter such as a cone radiating from
a spherical cap is applied, one finds that the depth at
which the indenter tip transitions from a spherical region of radius R to the conical aspect of the tip, h tip
occurs at
h tip
= 1 − sin θ
R
(16.5)
for a conical tip with an included half angle of θ.
As there are many analytical mechanics solutions developed for the contact of axisymmetric solids, one
391
Part B 16.1
While depth sensing indentation methods were being
developed, a change in imaging-based indention methods was also occurring. The development of the Vickers
(or diamond pyramid hardness) indentation test, and
determination of the angles used during pyramidal indentations [16.8], followed the basic method of the
Brinell test, but replaced the steel ball with a diamond
pyramid ground to an included angle between faces of
136◦ . It is interesting to note that the angle chosen was
based on the Brinell test. It had become common to
make indentations with the Brinell method to residual
indent diameters corresponding to 0.25–0.5 of the ball
diameter, and measurements of BHN as a function of
applied load showed an effective constant hardness region between d/D ratios of 0.25–0.5. The average of
these is 0.375, and if a four-sided pyramid is formed
around a spherical cap such that the ratio of the chordal
diameter to ball diameter is 0.375, the angle must be
136◦ . After applying a load, the indent is imaged, and
the diagonals of the indentation are measured. The
Vickers hardness is then defined as
◦
2P sin 136
2
(16.4)
DPH =
d2
16.1 Hardness Testing: Macroscopic Beginnings
392
Part B
Contact Methods
common technique is to apply an equivalent included
half angle for a cone that would provide the same relation between depth of contact and the projected contact
area of the solid. For both the Vickers and Berkovich
geometries, the ideal equivalent conical included half
angle is approximately 70.3◦ . For a tip radius of 200 nm,
the effective depth of transition is 12 nm.
A microhardness testing machine was modified to
continuously monitor the load and depth during an indentation by Loubet et al. [16.12] among others [16.13].
This led to the development of what is commonly
referred to as nanoindentation, though the term instrumented indentation is more accurate. Any procedure
which is capable of monitoring the load and depth during an indentation can be referred to as instrumented
indentation, continuous indentation or nanoindentation,
however inappropriate the scale may be when discussing indentations which penetrate depths of microns
and contact areas over 10 μm2 . The following section details the processes by which one experimentally
determines the properties of a material using these instrumented indentation techniques.
16.2 Extraction of Basic Materials Properties
from Instrumented Indentation
Part B 16.2
There are many recent reviews and resources which describe the general methods of determining the elastic
and plastic properties of materials using nanoindentation. This section will provide a brief review of the
developments in this area; readers that wish for a more
extensive treatment of the technique or specific aspects
are referred to several reviews [16.14–18]. In particular, this section will address the calculation of elastic
properties from an indentation load–depth curve, the
methods to determine hardness from the same test,
methods which have improved the technique, and address some of the limitations of the overall methodology
of nanoindentation.
16.2.1 General Behavior
of Depth Sensing Indentation
When a hard tip is impressed into a solid, the resulting load to the sample and resulting penetration of the
tip relative to the initial surface can be recorded during
the process. This is typically referred to as the load–
depth curve for a particular tip–solid system, and is
shown schematically in Fig. 16.1. The load–depth curve
has several important features which will be used in
the following discussion. The maximum load Pmax , the
maximum penetration depth h max , and the final depth h f
will be required to describe the indentation process. In
addition, the load–depth relationship can be determined
throughout the loading process, and the stiffness is
given by dP/ dh. As shown by Doerner and Nix [16.19],
and further developed by Oliver and Pharr [16.20],
an analysis of the unloading portion of the load–depth
curve allows the modulus and hardness of the material
to be determined; the following treatment is based on
that work. This type of analysis is based on the work
of Sneddon for the indentation of an elastic half space
with a rigid shape or column [16.21]. While Sneddon
originally used the analysis for the impression of an
axisymmetric punch, the current nanoindentation approach is often used for relatively shallow indenters of
pyramidal geometries such as the Berkovich geometry.
Under load, one assumes that the profile of the indentation is given by Fig. 16.2, where the contact radius (if
a conical indentation were being used) would be defined
by a, and the maximum penetration is given by h max .
Contact mechanics since the time of Hertz has recognized that the elastic deformation of two solids can be
Load P
Creep
Pmax
S = dP/dh
Loading
hf
Unloading
hmax
Displacement h
Fig. 16.1 Schematic load–depth curve for the penetration
of a indentation tip into a flat solid
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
described by composite terms. Extensive treatments of
contact mechanics for the advanced reader can be found
in Johnson’s text [16.22]. For the elastic compression of
two solids in contact, the compliances add in series, and
it is convenient to define a reduced elastic modulus E ∗
1 − νi2 1 − νs2
1
=
+
,
∗
E
Ei
Es
(16.6)
√
2
dP
= β √ E∗ A ,
dh
π
Initial
surface
(16.7)
a
Indenter
At Pmax
hc
hmax
Surface profile
After
unloading
Initial
surface
hf
Fig. 16.2 Schematic of the surface of a solid while in con-
tact with an indentation tip and after removing the loaded
tip
where A is the projected contact area of the solid
and β is a constant that attempts to account for the
differences between the axisymmetric contact models
and the experimental variations in using pyramidal geometries as well as possible variations due to plastic
deformation, and the fact that the contact area is beyond
the small-strain conditions assumed in many contact
mechanics problems. This correction factor β is still
a subject of debate and recent research and reported
values range from the commonly references King’s
1.034 [16.23] to values of over 1.1, as reviewed by
Oliver and Pharr [16.17] and Fischer-Cripps [16.24].
A recent study [16.25] describes these effects in detail.
The currently accepted recommendation is that using
β = 1.034 [16.17] or 1.05 [16.25] will allow reasonably
accurate values of elastic modulus to be determined,
given the other possible inaccuracies in the calibration
process.
There remain two system characteristics which must
be calibrated if an accurate assessment of the elastic
modulus and hardness of the sample are to be determined. Of course, the load–depth curve provides
measurements of P and h, but not A, which is needed to
calculate both the elastic properties and hardness of the
solid. First, one must estimate the contact depth from
the load–depth curve. The common method of assessing this requires the initial unloading slope, dP/ dh, to
be determined. If the indenter were a flat punch, then the
elastic unloading would be linear in load–depth space.
This was the initial assumption made by Doener and
Nix [16.19], that the initial unloading slope was linear.
However, if the contact was the elastic contact of a blunt
axisymmetric cone in the manner of Sneddon [16.26],
the relationship would be parabolic with depth. If the
contact were that of a sphere of radius R, rather than
a flat punch or cone, the original Hertzian elastic contact solution could be used, in which the relationship
would be to the 1.5 power. The loading conditions for
an axisymmetric elastic contact between an indenter and
a flat surface, such that a contact radius of a can be
described as
P = 2aE ∗ h
for a flat punch indenter of constant radius a,
2E ∗ 2
P=
h tan α
π
for a conical indenter of included half angle α,
4 P = E ∗ Rh 3
3
for a spherical indenter of radius R.
(16.8)
393
Part B 16.2
where E and v are the elastic modulus and Poisson’s ratio respectively, and the subscripts ‘i’ and ‘s’ refer to the
indenter and sample, respectively. For the conventional
indentation technique, diamond is selected for the indenter tip, and so 1141 GPa is commonly used for E i ,
while vi is 0.07.
During nanoindentation, one standard assumption
is that both elastic and plastic deformation occur during loading, beginning at the lowest measurable loads.
Section 16.5 will describe cases in which this is not
a logical assumption, but for many tests this assumption
will be valid. In this case, and with a shallow indenter (i. e., one with an included angle similar to that of
the Berkovich geometry), the unloading of the indentation will be used to determine the elastic properties
of the solid, since the common assumption is that the
unloading is purely described by the elastic response
of the system. In many metallic and ceramic systems
this is indeed experimentally observed (i. e., after partially unloading, subsequent reloading will follow the
load–depth curve until the previous maximum load is
achieved). If this is the case, then the unloading of the
system can be described by
Extraction of Basic Materials Properties
394
Part B
Contact Methods
Therefore, if the unloading slope, dP/ dh, is to be determined, it would at first glance appear to be possible
to select the appropriate relationship, perform a leastsquares fit on experimental data, and determine the
elastic modulus of the material. However, due to plastic
deformation, none of these methods are exactly accurate. In practice the easiest manner found to determine
the unloading slope is to perform a least-squares fit to
the unloading data using a power-law function which
can be described by
P = A(h − h f )m ,
(16.9)
Part B 16.2
where m will be between 1 and 2, and h f can be
estimated from the load–depth curve. This form is convenient in that the slope of this form will never be
negative during the unloading segment. In practice various portions of the unloading segment are selected for
fitting due to efforts to minimize drift or noise in the
system; hold periods are often inserted in the loading
schedule. Any curve fitting procedure should ignore regions of the unloading curve which contain extended
holding segments.
With the proper analysis of the unloading stiffness,
dP/ dh at the maximum depth and load, the contact
depth (Figs. 16.1 and 16.2) can be determined. For an
axisymmetric indenter the common method is to determine the contact depth h c as
hc = hf − ε
Pmax
,
dP/ dh
The frame compliance, commonly referred to in the
indentation literature as Cf , relates the displacement in
the system at a given applied load, wherein the system consists of the springs which suspend the indenter
tip, the shaft or screw which holds the indenter tip, and
the mounting mechanism for the sample. The displacement carried by the total system will be dependent on
each given sample due to mounting, but this is often
accounted for by determining the compliance for a typically stiff sample and then assuming that this frame
compliance is constant for other similarly mounted samples. The total load–displacement relationship can be
described by
dh
dh
=
+ Cf .
(16.11)
dP measured
dP sample
In practice, if the unloading stiffness is measured at
a variety of contact depths for a variety of stiff materials (sometimes fused silica, sapphire, and tungsten
are recommended), and once these values are inverted
this measured compliance is plotted as a function of
inverse contact depth, the intercept is Cf . This also allows the definition of the contact stiffness of the sample,
rather than of the system. The sample stiffness, S, is just
( dP/ dh)sample and will be used to determine the elastic
modulus from a given indentation.
16.2.2 Area Functions
(16.10)
where ε is a constant equal to 0.72 for an axisymmetric
indenter. Commonly ε is chosen to be 0.75, which describes the data well for pyramidal indenter geometries.
One last parameter must be determined experimentally to accurately determine the stiffness at unloading:
the frame compliance of the system itself. When any
mechanical test is performed, there will be a portion
of the displacement carried by the elastic deformation
of the loading frame. In bulk mechanical testing this is
well documented; if one wishes to utilize crosshead displacement and load data the compliance of the system is
subtracted at a given load, assuming a linear spring constant. In the case of bulk testing, the suggested method
to carry out mechanical testing is to utilize strain gauges
and extensometers rather than relying on the cross head
displacement, because the displacement given by the
frame may be a substantial part of the overall displacement. However, during nanoindentation testing there is
no direct analog to the extensometer, and therefore the
frame compliance must be accounted for during the test.
With all these parameters now defined, the final step
is to determine the area function of the tip. The initial point of analysis for this method relied upon the
contact area, not the depth of penetration, and therefore, in a manner reminiscent of the work of Rockwell,
the depth of penetration can be related to an area for
a given tip geometry. Many authors have developed
a series of techniques to relate the penetration depth to
the projected contact area of the tip with the sample.
Two popular methods are described in the literature;
one chooses an empirical functional dependence of
the projected contact area to contact depth, and the
other models the tip as a spherical cap on an effectively conical contact. In both cases, the calibration
begins by performing indentations to various depths in
materials of known elastic modulus (hence both are
sometimes referred to as constant-modulus methods).
The traditional materials chosen include fused silica and
single-crystal aluminum; however other materials such
as sapphire and tungsten are sometimes used as calibration standards. Rearranging (16.7), it is possible to
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
describe
A=
π
4
S
β E∗
2
(16.12)
and if E ∗ is constant, then A can be determined for any
given S at a contact depth h c . A can either be described
empirically [16.20] as
A = C1 h 2c + C2 h c + C3 h 0.5
c
+ C4 h 0.25
+ C51 h 0.125
+... ,
c
c
(16.13)
16.2.3 Assessment of Properties
During the Entire Loading Sequence
Analysis of the unloading slope provides information
from an individual point during an indentation, and as
much more data during an instrumented indentation
tests is collected, there has been an effort to develop
methods to extract more information about the properties of materials from the entire load–depth curve.
These fall into three categories, those which approach
the loading curve from an analytical standpoint, those
which impose a small oscillation upon the loading curve
395
to create a series of unloading segments, and those
which utilize finite element solutions to solve inverse
problems that provide more information about materials properties including strain hardening coefficients
and rate sensitivity of the test. The first two portions,
as they are still being used to primarily calculate E and
H, will be discussed here, while the third topic will be
covered in Sect. 16.3.
The first method to extract materials properties from
the entire loading segment focuses on the fact that
for a given material–tip combination the loading curve
tends to describe the balance between elastic and plastic deformation, referred to by Page as the mechanical
fingerprint of the material. Hainsworth and coworkers [16.27] developed a novel method initially proposed
by Loubet et al. [16.28] in which the shape of the loading curve instead of the unloading curve is considered.
This model is particularly applicable for evaluating materials where unloading curves exhibit a considerable
amount of elastic recovery. For instance, some coated
materials, such as CNx on silicon, show a well-behaved
loading curve but a highly curved unloading segment.
The relationship between loading displacement and
applied load was described by [16.27]
P = Kmh2 ,
(16.15)
where P is the applied load, h is the corresponding displacement, and K m is an empirical constant which is
a function of Young’s modulus, hardness as well as indenter geometry. For a Berkovich tip, K m is expressed
as
−2
E
H
.
(16.16)
K m = E 0.194
+ 0.930
H
E
Therefore, if either E or H is known, the other may
be calculated by curve fitting the experimental loading
curve using the combination of (16.16) and (16.15).
A second method to determine the properties of
materials during indentation builds upon the unloading
aspect of the test. If a sample is repeatedly loaded and
unloaded to greater depths, one effectively determines
the unloading slope at different contact depths. As the
initial unloading slope can be determined from a partially unloaded segment, the ability to sample in one
position the properties as a function of depth is particularly appealing in two cases: firstly where either the
material is layered or has some positional variations in
properties (or size-dependent properties) and secondly
when a spherical indentation is used for the experiment.
The instrumented indentation of a material with
a sphere would allow sampling at different contact-area-
Part B 16.2
where in this case C1 is often chosen to fit the ideal tip
shape (24.5 for a Berkovich tip). Subsequent constants
are empirically selected to provide a best fit to the elastic modulus from the measured data. The other method
assumes that the tip is described using a spherical cap
on an effective cone of included half angle α with a tip
radius of R [16.10]
h c C1 2
+
A=
C1 C2
πh 2c
+ 4Rπh c + 4R2 π cot2 α ,
=
(16.14)
cot2 α
where the constants C1 and C2 are related to α
and R, respectively. The other methodology used for
tip calibration relies upon actual measurement of the
tip as a function of position using scanning probe
microscopy (or some other quantitative profilometer
method) [16.11], and is less common due to the difficulty in performing these experiments; most users will
likely utilize the constant-modulus methods.
Having defined a method to determine the projected
contact area as a function of contact depth, the hardness H as determined in (16.2) is easily determined. The
hardness measured by nanoindentation is expected to
deviate from the hardness measured by post-indentation
inspection if the surface does not remain flat during the
entire test.
Extraction of Basic Materials Properties
396
Part B
Contact Methods
to-depth ratios, which is the same effect as measuring
hardness at different d/D ratios (see the previous
section). Field and Swain demonstrated this for nanoindentation using a cyclic loading–partial unloading
sequence [16.29]. By this time nanoindentation testing
was becoming more standardized, and they noted that
spherical indentations would probe different strains,
while conical or pyramidal indentation would probe
a fixed strain value; effectively (just like (16.3) but under load)
a
(16.17)
εi ≈ 0.2
R
for spheres while for pyramids and cones
εi ≈ 0.2 tan α ,
(16.18)
Part B 16.3
allowing them to calculate hardness as a function of
penetration depth. This paper also deals with the prediction of the load–depth graphs knowing materials
properties such as modulus and yield strength.
The last method of determining materials properties during nanoindentation is commonly referred to as
the continuous stiffness method, though other terms are
utilized by a variety of commercial manufacturers of
systems. In all cases, the process relies upon an imposed
oscillation in the force (or displacement) applied to the
tip. This is carried out at a specific frequency. The technique imposes a sinusoidal forcing function at a fixed
frequency ω, and uses this signal to calculate the contact
stiffness from
POS −1
2
S−1 + Cf
+ K s − mω2 + ω2 D2
h(ω) =
(16.19)
or from the phase difference between the displacement
and force signals from
ωD
,
(16.20)
tan(φ) = −1
−1
S + Cf
+ K s − mω2
where Cf is the compliance of the load frame, K S is
the stiffness of the column support springs, D is the
damping coefficient, POS is the magnitude of the forcing oscillation, h(ω) is the magnitude of the resulting
displacement oscillation, φ is the phase angle between
the force and displacement signals, and m is the mass.
Material properties can then be calculated at each of
these tiny unloading sequences. Further extension of
this work by Joslin and Oliver [16.30] led to the realization that, as both H and E ∗ use the area of contact
and applied load, at any point during the loading cycle
if the stiffness were measured using a lock-in amplifier
the quantity
P 4
(16.21)
S2 π
(neglecting correction factors such as β) provides
a measurement of the resistance to deformation which
does not require the use of an area function. For assessing the properties of materials which may be expected to
have depth-dependent hardnesses (for instance, in ionimplanted solids) this technique promised the ability
to assess these properties without the need to calibrate tips to extremely shallow depths. Of course, if
one knows the area function of the tip through a calibration technique, then it is possible to determine the
modulus directly if the stiffness is known at all points
during loading. The continuous stiffness method and its
variations are now common methods of assessing materials properties as a function of depth during a single
indentation. The obvious benefits of this method include the ability to sample small volumes (i. e., probe
laterally distinct features) which may not be able to
be the subject of many tests, to perform tip calibrations and property measurements in thin-film systems
rapidly, and to provide extremely sensitive methods of
system constants. For instance, a new method of determining frame compliance uses this technique[16.17];
when fused quartz is indented using the continuous stiffness method, P/S2 is a constant 0.0015 GPa−1 . This
method is quite sensitive to a system’s frame compliance, and obviously not to the area function. Therefore,
users of indentation systems can rapidly calibrate the
frame compliance to account for the variations in the
systems with time and ensure more accurate area functions for use in materials properties assessment.
H
2
E∗
=
16.3 Plastic Deformation at Indentations
The importance of imaging and examining indentations after they are made is often underestimated. While
there certainly is valuable data contained in load–depth
curves, there is also much information contained in the
plastically deformed region around the indentation, be
it in localized probes of bulk materials or in thin-film
coated systems [16.31]. Specifically, information about
the dislocation mechanisms responsible for deformation
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
can be obtained by examining atomic force microscopy
(AFM) images of the slip steps which develop on the
surface.
16.3.1 The Spherical Cavity Model
Θ
a
w
hf
c
Fig. 16.3 Definition of variables for indentations. Note that
the included angle Θ is only used here to represent the half
angle of the indentation. After the indenter tip has been
removed from the sample, the remaining angle in the residual cavity will increase due to the elastic recovery of the
indentation
included angle of the indenter is 2θ, the applied load is
P, and the extent of plastic deformation or plastic zone
radius is c. For this discussion the plastic zone will be
defined in the manner of Samuels and Mulhearn [16.33]
given as the radius of the edge of the vertical displacements of the surface.
Lockett [16.36] used a model of indentation which
relied on slip lines for included indenter angles between 105◦ and 160◦ . These calculations predict an
h/a ratio for a 105◦ indenter of 0.185, for a 120◦
indenter of 0.137, and for a 140◦ indenter of 0.085.
Dugdale [16.35] measured the pile up as a function
of indenter angle normalized by 2c, providing experimental data for steel, aluminum, and copper. When
converted to w/a, these values fall between approximately 0.2 and 0.1 for cone angles between 105◦ and
160◦ . The trend for Dugdale’s experimental data is that
the ratio c/a decreases as the indenter angle increases,
as do the calculations of Lockett. Mulhearn [16.37]
proposed that the maximum pile up depended on the
indenter angle, and as a first approximation the surface of the sample being indented would remain planar
during indentation. Upon removing the load, elastic
recovery would occur, and as an upper bound the maximum pile up could be approximated as 1/(6 tan θ).
Bower et al.’s model can be modified into a similar
ratio, and has a 1/ tan θ dependence, just as with the
model presented by Mulhearn. In addition, Bower’s
model suggests that the w/a ratio for a given indenter should be constant, independent of the applied
load.
The extent of plastic deformation around an indentation, often assumed to be linked to the lateral
dimensions of the out-of-plane deformation around the
indentation, is not the only way in which plastic zones
around indentations can be defined. In addition to the
upheaval around the indenter, there is a region centered
on the indenter tip which experiences radial compression from the indenter tip. The ratio of the plastic zone
size to the contact radius, c/a, will quantify the extent
of plastic deformation. According to Johnson’s conical spherical cavity model [16.32] of an elastic–plastic
material, the plastic zone is determined by
∗
1
E tan β
2 1 − 2ν 3
c
,
=
+
a
6σ y (1 − ν) 3 1 − ν
(16.22)
where α is the angle between the face of the cone
and the indented surface, 90 − θ and σy is the yield
strength of an elastic–plastic material. When the ratio
E ∗ tan α/σy is greater than 40, Johnson suggests that
397
Part B 16.3
The plastic deformation caused by an indentation has
two components, radial and tangential. The spherical
cavity model which has been developed extensively by
Johnson relies on this mode of deformation [16.32].
However, the spherical cavity model does not address
the amount of upheaval around an indentation, and
instead focuses on the radial expansion of the plastic zone. It has been shown [16.33] that materials
which exhibit perfectly plastic behavior tend to pile
up around an indenter, while annealed materials which
have high work hardening exponents tend to sink in
around an indenter. This behavior has been modeled
using finite element methods by Bower et al. [16.34].
Previous work on pile up around indentations has shown
the amount of pile up to vary with the indenter angle for a given load, with sharper indenters providing
more pile up [16.35]. This discussion will only consider cones between the angles of 105◦ and 137◦ ,
where the spherical cavity model has been used to
successfully determine plastic zones around indentations. Cones sharper than this may begin to approach
cutting mechanisms modeled by slip line fields, and
may be inappropriate for using a spherical cavity
model.
Figure 16.3 defines the variables which will be used
in this discussion. The maximum pile up is referred to
as w, the indentation depth relative to the nominal surface is h final , the contact radius of the indenter is a, the
16.3 Plastic Deformation at Indentations
398
Part B
Contact Methods
the analysis of elastic–plastic indentation is not valid,
and that the rigid–plastic case has been reached. Once
the fully plastic state has been reached, the ratio c/a
becomes about 2.33. Another model of the plastic zone
size [16.38] based upon Johnson’s analysis suggests that
the plastic zone boundary c is determined by
3P
.
(16.23)
c=
2πσ y
With this model, and as noted by Tabor, the hardness is
approximately three times the yield strength of a fully
plastic indentation and the ratio c/a should be 2.12
(similar to the suggested values of Johnson). In either
case, this provides a mechanistic reason for the conventional rules of indentation which require lateral spacing
of at least five times the indenter diameter to ensure no
influence of a previous indentation interferes with the
hardness measurement. Experimental measurements of
pile up around macroscopic indentations [16.39] have
shown the pile up models described in this section
to be reasonable, and both on the macroscopic and
nanoscale [16.40] the plastic zone size approximation
for the yield strength is reasonable.
Part B 16.3
16.3.2 Analysis of Slip Around Indentations
From what is known about the dislocation mechanisms
taking place beneath an indenter tip during indentation experiments, it seems a valid assumption that
dislocation cross-slip must take place for out-of-plane
deformation, or material pile up, on free surfaces around
indentations to occur [16.41–49]. Several studies on the
structure of residual impressions have been reported in
the literature. Kadijk et al. [16.50] has performed indentations in MnZn ferrite single crystals using spherical
tips and identified slip systems responsible for the patterns which resulted in the residual depression. Using
a combination of controlled etch pitting, chemomechanical polishing, and AFM, Gaillard et al. [16.46]
described dislocation structures beneath indentations
in MgO single crystals. Other work has been performed on body-centered cubic (BCC) materials to
identify changes in surface topography with crystal
orientation [16.48] as well as transmission electron microscopy (TEM) images of the subsurface dislocation
structure [16.49, 51]. Many of these results have noted
that the extent of plastic deformation may be dependent on the overall size of the indentation, as well as
exceeding the c/a ratio of 2.33 for nanoscale testing of
relatively defect-free materials.
The face-centered cubic (FCC) crystal structure contains only four unique slip planes. Chang
et al. [16.52] describe a method for determining the
surface orientation of a particular grain in an FCC material by measuring the angles of the slip step lines
on the surface around indentations and calculating the
orientation from the combination of angles. With the
current availability of orientation imaging microscopy
(OIM), it is possible to carry out a similar process in
reverse and use the known orientation of a grain to determine the slip plane responsible for each slip step.
Each slip plane can be indexed with respect to a reference direction taken from the OIM. Several studies
of surface topography have been performed [16.53, 54]
using this technique, and effects such as the include angle of the tip are shown to influence the extent of the
plastic zone around materials. Figure 16.4 demonstrates
the extent of surface deformation in a stainless-steel alloy indented with a spherical tip with a tip radius of
approximately 10 μm, and a 90◦ conical indenter with
a tip radius of approximately 1 μm. For the second indentation the extent of plastic deformation surpasses the
expected c/a ratio of 2.33. These results are not unique
to surface measurements; dislocation arrays identified
by etching after indentation in MgO have demonstrated
similar effects [16.44]. Clearly the extent of deformation in the lateral direction which is transferred to the
surface is influenced by the effective strain of the indentation. Of particular interest in recent studies using
dislocation etch pitting in conjunction with nanoinden-
20 μm
20 μm
Fig. 16.4 Slip steps from only the positively inclined slip
planes around blunt tip indentations in a grain with surface orientation of (1 2 30). When the effective strain is
increased by using a sharp tip, steps form from both the
positively and negatively inclined slip planes
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
tation experiments [16.55] is that the lattice friction
stress for individual screw and edge dislocation components can be measured; Gaillard and co-workers
16.4 Measurement of Fracture Using Indentation
399
have determined lattice friction stresses of 65 MPa for
edge dislocations and 86 MPa for screw dislocations in
MgO.
16.4 Measurement of Fracture Using Indentation
If nanoindentation is to prove as useful as bulk mechanical tests, properties other than hardness (which is poorly
defined in a mechanics sense) and elastic modulus must
be considered. Some of these properties, such as yield
strength and strain hardening, were covered in the previous section. Two other main areas which provide data
to be utilized in assessing a material’s response to an
applied load are time-dependent deformation and mechanisms of fracture. In the case of fracture measurements
observation of the surface of the material will generally
be required.
16.4.1 Fracture Around Vickers Impressions
where c is the crack radius, P is the peak indentation
load, and χr is an dimensionless constant dependent
on the specific indenter–material system that can be
found by applying indentations to various maximum
loads and measuring crack lengths. If the material system has a constant toughness, then a fitting procedure
can be used to determine the dimensionless constant.
15 μm
Fig. 16.5 Vickers indentation in silicon, showing radical
cracks
Part B 16.4
The indentation fracture technique is widely used to
characterize the mechanical behavior of brittle materials due to its simplicity and economy in data
collection [16.56]. The fracture patterns strongly depend on loading conditions, which fall into two basic
categories: blunt or sharp contact. While blunt indenters (for instance, spherical tips) are associated with
a Hertzian elastic stress field, sharp indenters (for instance Vickers or Knoop) lead to an elastic–plastic
field underneath the contact region [16.57]. In the later
case, radial cracks may be generated from the corners of the contact impression, and median cracks
propagate parallel to the load axis beneath the plastic deformation zone in the form of circular segments
truncated by the material surface. At higher peak loads,
lateral cracks may also be formed in the manner of
a saucer-shaped surface centered near the impression
base [16.57]. As the driving force for lateral crack
growth is rather complex compared to the centerloaded symmetric radial/median stress field, for the
sake of simplicity, only radial/median crack system is
considered in terms of indentation fracture mechanics
herein.
Lawn and coworkers developed a model to quantitatively represent the relationship of the radial crack
size and the fracture toughness for a ceramic material [16.58]. The key idea was to divide the indentation
stress field into elastic and residual components where
the reversible elastic field enhances median extension
during the loading half cycle and the irreversible plastic field primarily provides the driving force for radial
crack evolution in the unloading stage. In a later study,
they further the discussion by considering the effect
of a uniform biaxial applied field on the crack evolution [16.59, 60] based on experimental observation that
an imposed uniform stress field could alter the fracture
behavior by either increasing or decreasing the crack
size [16.61, 62]. In general, the toughness of a material
is related to the extent of radial cracks that emanate from
the corners of an indentation, such as those shown in
Fig. 16.5. An extensive review of the modes of fracture
can be found in the literature [16.57]. For the commonly reported fracture tests using Vickers indentation,
the stress intensity factor due to the residual impression
is
0.5
E
P
P
= χr 1.5 ,
(16.24)
K r = ζr
H
c1.5
c
400
Part B
Contact Methods
tests have arrested, the assumption that K r is equal to the
fracture toughness is used to provide a measurement of
the toughness of the system.
Load P (mN)
600
500
16.4.2 Fracture Observations
During Instrumented Indentation
400
300
Pop-in
200
hf
100
0
0
1000
2000
3000
4000
5000
Displacement h (nm)
Fig. 16.6 Nanoindentation in fused silica using a 42◦ in-
denter
Of course, if a different material is tested one must
determine the constant again and ζr is often used instead, where ζr is given as 0.016 [16.61] or sometimes
0.022 [16.63]. Since the cracks measured during these
Part B 16.4
a) Load P (mN)
There are drawbacks to using static indentation for
fracture. First, many materials which are brittle do
not exhibit fracture during nanoindentation using conventional indentation probes. Secondly, imaging the
indentation can become problematic if very small features or indentations are made in the sample. Therefore,
it would be convenient to determine the occurrence
of fracture during an instrumented indentation. Some
authors have chosen to identify fracture mechanisms
using acoustic emission sensors during nanoindentation [16.64–66], but this does require specialized
equipment and analysis to separate fracture from plasticity events. Morris et al. [16.67] have utilized sharper
indenter probes to determine the effects of included
angle on the resulting load–displacement curves, as
shown in the typical result in Fig. 16.6. More sensitive measurements of these pop-in events are found if
one measures the stiffness continually during the ex-
b) Load P (mN)
600
600
500
500
52°
400
400
65°
300
65°
52°
42°
35°
300
42°
200
200
35°
100
0
c)
100
0
100
200
300
400
500
0
Displacement h (nm)
0
100
200
300
400
500
Displacement h (nm)
d)
Fig. 16.7a–d Unloading curves and
10 μm
respective images of residual indentation impressions showing cracking
an systems ((a) and (b)), while uniform unloading curves occur in
materials which undergo only plastic
deformation ((c) and (d))
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
16.4 Measurement of Fracture Using Indentation
401
Load (μN)
3000
Lower than the excursion load
Higher than the excursion load
fim thickness: 92 nm
2500
2000
1500
Cracks in oxide film
1000
1000 nm
500
Fig. 16.8 AFM image of circumferential cracks around
indentations into anodized titanium. The large ring corresponds to a through-thickness crack, while the smaller
impression in the center is the contact area defined by the
indenter tip [16.68]
0
20
40
60
80
100
120
140
Depth (nm)
Fig. 16.9 Indentation into a 92 nm-thick TiO2 film grown
anodically on a Ti substrate. The distinct excursion in
depth, in this case corresponds to a through-thickness fracture in the film [16.68]
to produce cracks in the film [16.75–80]. Malzbender
and de With [16.75] demonstrated that the dissipated
energy was related to the fracture toughness of the
coating and interface by performing simple integrations
of the loading and unloading portions from indentation to determine the amount of energy needed to
damage the film. Other studies have suggested that
the fracture energy could be estimated as the energy
consumed during the first circumferential crack during the load drop or plateau on the load–displacement
curve [16.74, 76, 78, 81]. A recent study has demonstrated one technique which relies upon examination
of the load–depth curve, knowing the general behavior of hard film–soft substrate systems [16.68, 82]. In
this case, the load–depth curve of a coated system,
shown in Fig. 16.9 for a 90 nm TiO2 on Ti, also demonstrates a pop-in or excursion in the load–depth curve.
These pop-in events, present only because the indentation testing system is a load-controlled device, are
in this case indicative of film fracture because permanent deformation is observed prior to the pop-in. The
plastic zone c from Sect. 16.3 is used to estimate the radius of the fracture, and the toughness of the film can
be estimated from the energy difference between that
which goes into the indentation of the film–substrate
system and indentations to similar depths for the substrate only.
Part B 16.4
periment. While it may be tempting to identify the
presence of a pop-in to a fracture event, these authors
demonstrate conditions under which no pop-in event occurs and yet the resulting indentation impression instead
shows clear evidence of cracking. Probes of increasing acuity (moving from the conventional Berkovich
geometry to sharper cube-corner-type systems) are
more likely to generate fracture. A comparison of
the unloading slopes of probes of various acutities
demonstrated that samples which exhibit indentationinduced fracture demonstrate differences in unloading
(while having the same final displacement), while materials which only exhibit plastic deformation do not
show differences in unloading curves, as shown in
Fig. 16.7. Therefore, the authors suggest utilizing a shallow (Berkovich) and acute (cube corner or similar)
indenter pair, and that comparing the ratio of h max /h f
for the different tips will enable the user to identify
materials which are susceptible to indentation-induced
fracture.
Another case for fracture measurements particularly
suited to continuous indentation is the application of
hard films on compliant or soft substrates, which results in circumferential fracture around the indentation,
as shown in Fig. 16.8. Thin films have been shown to
fracture during indentation processes at critical loads
and depths [16.69]. Some studies have analyzed these
fracture events by calculating an applied radial stress
at fracture [16.68, 70, 71], an applied stress intensity at
fracture [16.71–74], and the amount of energy required
0
402
Part B
Contact Methods
16.5 Probing Small Volumes
to Determine Fundamental Deformation Mechanisms
Part B 16.5
It has been well documented that small volumes of materials regularly exhibit mechanical strengths greatly
in excess of more macroscopic volumes. Observations
in whiskers of metals were among the first studies to
demonstrate the ability of metals to sustain stresses
approaching the theoretical shear strength of the material. The often described indentation size effect in
metals [16.83–86] has been the subject of many studies with nanoindentation due to the ability to neglect
elastic recovery (i. e., the indentation hardness is measured under load) and to measure very small sizes. In
ceramics this effect has been explained by variations in
elastic recovery during the indentation [16.87]. Results
on nanocrystalline metals have continued that trend,
until the literature has reached a point where models
suggest that grain sizes too small to support the nucleation and growth of stable dislocation loops will
likely deform via other mechanisms. Similarly, nanostructured metallic laminates have been demonstrated to
exhibit particularly unique deformation behavior. Other
recent experiments in which a flat punch in a nanoindenter has been used to carry out compression tests on
machined structures fabricated via focus ion beam machining have demonstrated that just having a smaller
volume of material, with the concomitant increase in
surface area and decrease in sample size, may indeed impact plasticity in ways not immediately obvious
through scaling arguments [16.88, 89]. Recent work
has also demonstrated the ability to quantify the stress
required to cause the onset of plasticity dislocations
in relatively dislocation-free solids [16.90–94]. This
method is only possible using small-scale mechanical
testing; large-volume mechanical testing will generally measure the motion of pre-existing dislocations
under applied stresses. Nanoindentation couples well
with small-scale mechanical modeling, as it approaches
volumes which can be simulated using molecular
dynamics [16.95, 96] and the embedded atom technique [16.97].
In materials with low dislocation densities, the sudden onset of plasticity occurs at stresses approaching
the theoretical strength of a material. Recent studies
have proposed two primary models of these effects:
homogenous nucleation and thermally activated processes. Gane and Bowden were the first to observe
the excursion phenomena on an electropolished surface of gold [16.98]. A fine tip was pressed on the
gold surface while observing the deformation in a scan-
ning electron microscope. No permanent penetration
was observed initially, but at some point during the process a sudden jump in displacement occurred, which
corresponded to the onset of permanent deformation.
Similar observations of nonuniform loading during indentation were made by Pethica and Tabor [16.99],
who applied a load to bring a clean tungsten tip into
contact with a nickel (111) surface in an ultrahighvacuum environment while monitoring the electrical
resistance between the tip and surface. The nickel had
been annealed, sputtered to remove contaminants, and
then annealed again to remove surface damage from the
sputtering process. When an oxide was grown on the
nickel to thicknesses of greater than 50 Å, loading was
largely elastic while the electrical resistance remained
extremely high, with only minimal evidence of plastic
deformation. However, past some critical load the resistance between the tip and sample dropped dramatically,
with continued loading suggestive of largely plastic deformation.
With the advent of instrumented indentation testing
it was possible to monitor the penetration of an indenter tip into a sample. The pop-in or excursion in depth
during indentation was observed by many researchers
during this period, and is represented in Fig. 16.10.
Load (mN)
6
5
Elastic loading
prediction
4
3
2
1
0
0
50
100
150
200
Depth (nm)
Fig. 16.10 Nanoindentation into electropolished tungsten,
with an initial elastic loading followed by plastic deformation. Elastic loading prediction by [16.26]
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
Some researchers have utilized experimental methods
that are either depth controlled [16.100,101] or a system
in which the overall displacement (and not penetration)
was held constant [16.91, 102]. Post-indentation microscopy demonstrated that, after these discontinuous
events during loading in single-crystal ceramics, dislocation structures were present [16.49, 55]. Using the
Hertzian elastic model,
4 (16.25)
P = E ∗ Rδ3 ,
3
the maximum shear stress under the indenter tip τmax is
related to the mean pressure pm and is given by
1
6(E ∗ )2 3 1/3
P
(16.26)
τmax = 0.31
π 3 R2
at a position underneath the indenter tip at a depth of
0.48a, where a is the contact area, and is determined by
solving
∗2 1/3
6E
3P
=
P 1/3 .
(16.27)
2πa2
π 3 R2
Recently, Schuh and co-workers [16.106] carried
out indentation experiments that probed the onset of
plastic deformation at elevated temperatures to determine an effective activation energy and volume for the
initiation of plasticity. Using a method describing the
probability of an excursion occurring underneath the indenter tip they use an Arrhenius model for dislocation
nucleation. They suggest that the activation energy is
similar to that of vacancy motion, and that the activation volume is on the order of a cubic Burgers vector.
This suggests that point defects have a direct relationship with dislocation nucleation. Another recent study
of incipient plasticity in Ni3 Al [16.107] suggests that
the growth of a Frank dislocation source at subcritical stresses controls plasticity at lower stresses during
indentation, and that self-diffusion along the indenter–
sample surface dominates the onset of plasticity in these
cases of nanoindentation.
Bahr and Vasquez [16.108] have shown that, for
solid solutions of Ni and Cu, the shear stress under
the indenter tip at the point of dislocation nucleation
can be correlated to changes in the elastic modulus of
the material. Using a diffusion couple of Ni and Cu
to create a region of material with a range of compositions ranging from pure Ni to pure Cu, it was
shown that the changes in shear stress required to nucleate dislocations are on the order of the changes in
elastic modulus, between 1/30 and 1/20 of the shear
modulus of this alloy. The implication of these data
is that overall dislocation nucleation during nanoindentation is strongly related to shear modulus in this
system, but does not preclude the diffusion mechanism suggested by Ngan and Po [16.107]. Another
example of these methods is nanoindentation being
used to probe the effects of hydrogen on dislocation
nucleation and motion in a stainless steel [16.109].
These pop-in events were used to demonstrate that,
while dissolved hydrogen lowers the indentation load
at which dislocations are nucleated, this was likely a result of hydrogen decreasing the shear modulus and not
directly related to hydrogen creating or modifying nucleation sites. The pop-ins occurred more slowly in
the presence of hydrogen, most likely due to hydrogen
inhibiting the mobility of fast-moving dislocations as
they are rapidly emitted from a source. Finally, a second excursion occurs after charging with hydrogen, but
is uncommon in uncharged material. When coupled
to analysis of slip steps around the indentations that
show the existence of increased slip planarity, this suggests that dislocations emitted during the first excursion
are inhibited in their ability to cross-slip and there-
403
Part B 16.5
The shear stress field does not drop off that rapidly,
and there is a region under the indenter tip extending
to approximately 2a which has an applied shear stress
of approximately 0.25 pm . Many research groups have
noted that this stress approaches the theoretical shear
stress in a crystal. The similarity between the applied
shear stress and shear modulus is close to the classical models of homogeneous dislocation nucleation
espoused by the early theoretical work on dislocation
mechanisms. For instance, Cotrell noted that the classical theoretical shear strength of a material is often
approximated by μ/30 to μ/100 [16.103], where μ is
the shear modulus.
A new set of experiments utilizing in situ nanoindentation in a transmission electron microscope has
been used to support this model of homogeneous
dislocation nucleation. Minor et al. [16.92] observed
dislocations to pop out underneath an indenter tip after some initial elastic loading during contact between
the tip and aluminum sample. This direct observation
is strong evidence that dislocations nucleate underneath
the indenter tip. However, the most recent studies from
in situ microscopy have shown that even the very initial contact can generate dislocations that nucleate and
propagate to grain boundaries, after which an elastic
loading behavior is again demonstrated followed by
a second strain burst [16.104]. Multiple yield points during nanoindentation are often referred to as staircase
yielding [16.93, 101, 105].
Probing Small Volumes
404
Part B
Contact Methods
fore cannot accommodate a fully evolved plastic zone.
A second nucleation event is then needed on a differ-
ent slip system to accommodate the plasticity from the
indenter.
16.6 Summary
This chapter has provided a summary of the basic
methodologies for assessing the mechanical properties of materials on a small scale via instrumented
indentation, often more commonly referred to as
nanoindentation. The first two sections described the
history of indentation testing and the concept of
hardness, beginning with the quantification carried
out by Brinnel. This aimed to provide a background for the reader, who may feel at times that
many of the conventions in nanoindentation appear
to be based on somewhat arbitrary rules. Moving
to instrumented techniques, the development of contact mechanics approaches for determining elastic
and plastic properties of solids via small-scale indentations was outlined, including the analysis of
the unloading slope and the imposed oscillation
techniques currently available on commercial instruments.
The later sections of the chapter focused on
more advanced uses of small-scale indentations. The
deformation surrounding an indentation provides a researcher with more information than just the elastic
properties; sophisticated analysis methods allow for
determination of more complex mechanical properties
such as strain hardening. Indentation-induced fracture,
while still a subject of debate in the literature, is often
used in brittle materials, so this topic is covered herein
to provide the reader with a current status of the methods. Finally, the last section covers more fundamental
dislocation behavior (such as nucleation or multiplication issues) that is often examined by very low-load
indentations.
Part B 16
References
16.1
16.2
16.3
16.4
16.5
16.6
16.7
16.8
16.9
16.10
16.11
H. Hertz: Miscellaneous Papers. English translation
by D.E. Jones, G.E. Schott (MacMillan, New York,
1896) pp. 163-183
F. Auerbach: Smithsonian Report for 1891 (Government Printing Office, Washington 1893) pp. 207–236
J.A. Brinnnel: II.Cong. Int. Methodes d’Essai, J. Iron
Steel Inst. 59, 243 (1901), translated to English by
A. Wahlberg
E. Meyer: Untersuchungen über Härteprüfung und
Härte, Z. Ver. Dtsch. Ing. 52, 645–654 (1908)
S.L. Hoyt: The ball indentation hardness test, Trans.
Am. Soc. Steel Treating, 6, 396 (1924)
D. Tabor: The Hardness of Metals (Oxford Univ.
Press, Oxford 1951)
V.E. Lysaght: Indentation Hardness Testing (Reinhold Pub., New York 1949)
R.L. Smith, G.E. Sandland: Some notes on the use
of a diamond pyramid for hardness testing, J. Iron.
Steel Inst. 3, 285–301 (1925)
E.S. Berkovich: Three faceted diamond pyramid for
micro hardness testing, Industrial Diamond Rev. 11,
129–133 (1951)
J. Thurn, R.F. Cook: Simplified area function for
sharp indenter tips in depth-sensing indentation,
J. Mater. Res. 17, 1143–1146 (2002)
M.R. Vanlandingham, T.F. Juliano, M.J. Hagon:
Measuring tip shape for instrumented indentation
16.12
16.13
16.14
16.15
16.16
16.17
16.18
16.19
using atomic force microscopy, Measurement Sci.
Tech. 16, 2173–2185 (2005)
J.L. Loubet, J.M. Georges, O. Marchesini, G. Meille:
Vickers indentation curves of magnesium oxide
(MgO), J. Tribology, 106, 43–48 (1984)
D. Neweyt, M.A. Wilkins, H.M. Pollock: An ultralow-load penetration hardness tester, J. Phys. E:
Sci. Instrum., 15, 119–122 (1982)
A.C. Fischer-Cripps: Nanoindentation, 2nd edn.
(Springer, Berlin Heidelberg 2004)
B. Bhushan (Ed.): Springer Handbook of Nanotechnology, 2nd edn. (Springer, Berlin Heidelberg
2004)
X. Li, B. Bhushan: A review of nanoindentation
continuous stiffness measurement technique and
its applications, Mater.Char. 48, 11–36 (2002)
W.C. Oliver, G.M. Pharr: Review: Measurement of
hardness and elastic modulus by instrumented
indentation: Advances in understanding and refinements to methodology, J. Mater. Res. 19, 3–20
(2004)
M.R. VanLandingham: Review of instrumented indentation, J. Res. Natl. Inst. Stand. Technol. 108,
249–265 (2003)
M.F. Doerner, W.D. Nix: A method for interpreting
the data from depth-sensing indentation instruments, J. Mater. Res. 1, 601–609 (1986)
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
16.20
16.21
16.22
16.23
16.24
16.25
16.26
16.27
16.28
16.30
16.31
16.32
16.33
16.34
16.35
16.36
16.37
16.38
16.39
16.40
16.41
16.42
16.43
16.44
16.45
16.46
16.47
16.48
16.49
16.50
16.51
S. Harvey, H. Huang, S. Vennkataraman, W.W. Gerberich: Microscopy and microindentation mechanics of single crystal Fe-3 wt.% Si: Part I. Atomic
force microscopy of a small indentation, J. Mater.
Res. 8, 1291–1299 (1993)
D.F. Bahr, W.W. Gerberich: Pile up and plastic zone
size around indentations, Metall. Mater. Trans. A
27A, 3793–3800 (1996)
D. Kramer, H. Huang, M. Kriese, J. Robach, J. Nelson, A. Wright, D. Bahr, W.W. Gerberich: Yield
strength predictions from the plastic zone around
nanocontacts, Acta Mater. 47, 333–343 (1998)
Y. Gaillard, C. Tromas, J. Woirgard: Pop-in
phenomenon in MgO and LiF: observation of dislocation structures, Phil. Mag. Leters 83, 553–561
(2003)
E. Carrasco, M.A. Gonzalez, O. Rodriguez de la
Fuente, J.M. Rojo: Analysis at atomic level of
dislocation emission and motion around nanoindentations in gold, Surf. Sci. 572, 467–475 (2004)
M. Oden, H. Ljungcrantz, L. Hultman: Characterization of the induced plastic zone in a single crystal
TiN(001) film by nanoindentation and transmission
electron microscopy, J. Mater. Res. 12, 2134–2142
(1997)
C. Tromas, J. C. Girard, V. Audurier, J. Woirgard,
Study of the low stress plasticity in single-crystal
MgO by nanoindentation and atomic force microscopy, J. Mater. Sci. 34, 5337–5342 (1999)
W. Zielinski, H. Huang, W.W. Gerberich: Microscopy
and microindentation mechanics of single crystal
Fe-3 wt.%Si: Part II. TEM of the indentation plastic
zone, J. Mater. Res. 8, 1300–1310 (1993)
Y. Gaillard, C. Tromas, J. Woirgard: Study of
the dislocation structure involved in a nanoindentation test by atomic force microscopy and
controlled chemical etching, Acta Mater. 51, 1059–
1065 (2003)
P. Peralta, R. Ledoux, R. Dickerson, M. Hakik,
P. Dickerson: Characterization of surface deformation around Vickers indents in monocrystalline
materials, Metall. Mater. Trans. A 35A, 2247–2255
(2004)
N.A. Stelmashenko, M.G. Walls, L.M. Brown,
Y.U.V. Milman: Microindentations on W and Mo
oriented single crystals: An STM study, Acta Metal.
Mater. 41, 2855–2865 (1993)
T.F. Page, W.C. Oliver, C.J. McHargue: Deformation
behavior of ceramic crystals subjected to very low
load (nano)indentations, J. Mater. Res. 7, 450–473
(1992)
S.E. Kadijk, A. Broese van Groenou: Cross-slip patterns by sphere indentations on single crystal MnZn
ferrite, Acta Metall. 37, 2625–2634 (1989)
W. Zielinski, H. Huang, S. Venkataraman,
W.W. Gerberich: Dislocation distribution under
a microindentation into an iron-silicon single
crystal, Phil. Mag. A 72, 1221–1237 (1995)
405
Part B 16
16.29
W.C. Oliver, G.M. Pharr: An improved technique
for determining hardness and elastic modulus using load and displacement sensing indentation
experiments, J. Mater. Res. 7, 1564–1583 (1992)
I.N. Sneddon: The relation between load and penetration in the axisymmetric Boussinesq problem
for a punch of Arbitrary Profile, Int. J. Engng. Sci.
3, 47–56 (1965)
K.L. Johnson: Contact Mechanics (Cambridge Univ.
Press, Cambrige 1985)
R.B. King: Elastic analysis of some punch problems
for a layered medium, Int. J. Solids Structures 23,
1657–1664 (1987)
A.C. Fischer-Cripps: Critical review of analysis and
interpretation of nanoindentation test data, Surf.
Coatings Tech. 200, 4153–4165 (2006)
J.H. Strader, S. Shim, H. Bei, W.C. Oliver, G.M. Pharr:
An experimental evaluation of the constant β relating the contact stiffness to the contact area
in nanoindentation, Phil. Mag. 86, 5285–5298
(2006)
I.N. Sneddon: Boussinesq’s problem for a rigid
cone, Proc. Cambridge Phil. Soc. 44, 492–507 (1948)
S.V. Hainsworth, H.W. Chandler, T.F. Page: Analysis of nanoindentation load-displacement loading
curves, J. Materials Research 11, 1987–1995 (1996)
J.L. Loubet, J.M. Georges, J. Meille: Microindentation Techniques in Materials Science and
Engineering (American Society for Testing and Materials, Philadelphia 1986), ed. by P. J. Blau, B. R.
Lawn, Vol. 72
J.S. Field, M.V. Swain: A simple predictive model
for spherical indentation, J. Mater. Res. 8, 297–306
(1993)
D.L. Joslin, W.C. Oliver: A new method for analyzing
data from continuous depth-sensing microindentation tests, J. Mater. Res. 5, 123–126 (1990)
T.F. Page, S.V. Hainsworth: Using nanoindentation
techniques for the characterization of coated systems: a critique, Surf. Coating Technol. 61, 201–208
(1993)
K.L. Johnson: The correlation of indentation experiments, J. Mech. Phys. Solids 18, 115–126 (1970)
L.E. Samuels, T.O. Mulhearn: An experimental investigation of the deformed zone associated with
indentation hardness impressions, J. Mech. Phys.
Solids 5, 125–134 (1957)
A.F. Bower, N.A. Fleck, A. Needleman, N. Ogbonna:
Indentation of a Power Law Creeping Solid, Proc.
Royal. Soc. 441A, 97–124 (1993)
D.S. Dugdale: Cone indentation experiments, J.
Mech. Phys. Solids, 2, 265–277 (1954)
F.J. Lockett: Indentation of a rigid/plastic material
by a conical indenter, J. Mech. Phys. Solids 11, 345–
355 (1963)
T.O. Mulhearn: The Deformation of Metals by
Vickers-Type Pyramidal Indenters, J. Mech. Phys.
Solids 7, 85–96 (1959)
References
406
Part B
Contact Methods
16.52
16.53
16.54
16.55
16.56
16.57
16.58
16.59
16.60
Part B 16
16.61
16.62
16.63
16.64
16.65
16.66
16.67
S.C. Chang, H.C. Chen: The determination of F.C.C.
crystal orientation by indentation, Acta Metall
Mater. 43, 2501–2505 (1995)
K.A. Nibur, D.F. Bahr: Identifying slip systems
around indentations in FCC metals, Scripta Materialia 49, 1055–1060 (2003)
K.A. Nibur, F. Akasheh, D.F. Bahr: Analysis of dislocation mechanisms around indentations through
slip step observations, J. Mater. Sci. 42, 889–900
(2007)
Y. Gaillard, C. Tromas, J. Woirgard: Quantitative
analysis of dislocation pile-ups nucleated during
nanoindentation in MgO, Acta Mater. 54, 1409–1417
(2006)
B.R. Lawn, T.R. Wilshaw: Fracture of Brittle Solids
(Cambridge Univ. Press, Cambridge 1975)
R.F. Cook, G.M. Pharr: Direct observation and
analysis of indentation cracking in glasses and
ceramics, J. Am. Cer. Soc. 73, 787–817 (1990)
B.R. Lawn, A.G. Evans, D.B. Marshall: Elastic/Plastic Indentation Damage In Ceramics: The
Median/Radial Crack System, J. Am. Cer. Soc. 63,
574–581 (1980)
D.B. Marshall, B.R. Lawn: J. Mater. Sci., Residual
stress effects in sharp contact cracking Part 1, Indentation fracture mechanics 14, 2001–2012 (1979)
D.B. Marshall, B.R. Lawn, P. Chantikul: Residual
stress effects in sharp contact cracking, Part 2.
Strength degradation, J. Mater. Sci. 14, 2225–2235
(1979)
G.R. Anstis, P. Chantikul, B.R. Lawn, D.B. Marshall:
A critical evaluation of indentation techniques
for measuring fracture toughness I, Direct crack
measurements, J. Am. Ceram. Soc. 64, 533–542
(1981)
P. Chantikul, G.R. Anstis, B.R. Lawn, D.B. Marshall:
A critical evaluation of indentation techniques for
measuring fracture toughness II, Strength method,
J. Am. Ceram. Soc. 64, 539–543 (1981)
R.F. Cook: Strength Characterization of Ceramics
using Controlled Indentation Flaws. Ph.D. Thesis
(Univ. New South Wales, Australia 1986)
W.W. Gerberich, W.M. Mook, M.J. Cordill,
J.M. Jungk, B. Boyce, T. Friedmann, N.R. Moody,
D. Yang: Nanoprobing fracture length scales, Int. J.
Fract. 138, 75–100 (2006)
D.F. Bahr, J.W. Hoehn, N.R. Moody, W.W. Gerberich: Adhesion and acoustic emission analysis
of failures in nitride films with a metal interlayer,
Acta Mater. 45, 5163–5175 (1997)
A. Daugela, H. Kutomi, T.J. Wyrobeck: Nanoindentation induced acoustic emission monitoring of
native oxide fracture and phase transformations,
Z. Metallkunde 92, 1052–1056 (2001)
D.J. Morris, S.B. Myers, R.F. Cook: Sharp probes
of varying acuity: Instrumented indentation and
fracture behavior, J. Mater. Res. 19, 165–175
(2004)
16.68
16.69
16.70
16.71
16.72
16.73
16.74
16.75
16.76
16.77
16.78
16.79
16.80
16.81
16.82
16.83
M. Pang, D.F. Bahr: Thin film fracture during
nanoindentation of a titanium oxide film – titanium system, J. Mater. Res. 16, 2634–2643 (2001)
N.G. Chechenin, J. Bottiger, J.P. Krog: Nanoindentation of amorphous aluminum oxide films II.
Critical parameters for the breakthrough and a
membrane effect in thin hard films on soft substrates, Thin Solid Films 261, 228–235 (1995)
D.F. Bahr, C.L. Woodcock, M. Pang, K.D. Weaver,
N.R. Moody: Indentation induced film fracture in
hard film – soft substrate systems, Inter. J. Frac.
119/120, 339–349 (2003)
E. Wepplemann, M.V. Swain: Investigation of the
stresses and stress intensity factors responsible for
fracture of thin protective films during ultra-micro
indentation tests with spherical indenters, Thin
Solid Films 286, 111–121 (1996)
J. Thurn, R.F. Cook: Mechanical and thermal properties of physical vapour deposited alumina films
Part II Elastic, plastic, fracture, and adhesive behaviour, J. Mater. Sci. 39, 4809–4819 (2004)
C. Herzl: Fracture mechanics analysis of thin coatings under plane-strain indentation, Int. J. Sol.
Struct. 40, 591–610 (2003)
J. Malzbender, G. de With: Elastic modulus, indentation pressure and fracture toughness of hybrid
coatings on glass, Thin Solid Films 366, 139–149
(2000)
J. Malzbender, G. de With: Energy dissipation,
fracture toughness and the indentation load–
displacement curve of coated materials, Surf. Coat.
Tech. 135, 60–68 (2000)
P.G.T. van der Varst, G. de With: Energy based approach to the failure of brittle coatings on metallic
substrates, Thin Solid Films 384, 85–91 (2001)
A. Abdul-Baqi, E. Van der Giessen: Numerical analysis of indentation-induced cracking of brittle
coatings on ductile substrates, Int. J. of Sol. Struct.
39, 1427–1442 (2002)
X. Li, D. Diao, B. Bhushan: Fracture mechanisms of
thin amorphous carbon films in nanoindentation,
Acta Mater. 45, 4453–4461 (1997)
K. Sriram, R. Narasimhan, S. Biswas: A numerical
fracture analysis of indentation into thin hard films
on soft substrates, Eng. Fract. Mech. 70, 1323–1338
(2003)
Z. Xia, W. Curtin, B. Sheldon: A new method to
evaluate the fracture toughness of thin films, Acta.
Mater. 52, 3507–3517 (2004)
X. Li, B. Bhushan: Measurement of fracture toughness of ultra-thin amorphous carbon films, Thin
Solid Films 315, 214–221 (1998)
K.R. Morasch, D.F. Bahr: An energy method to analyze through thickness thin film fracture during
indentation, Thin Solid Films 515, 3298–3304 (2007)
N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson: Strain gradient plasticity: Theory and experiment, Acta Metall. 42, 475–487 (1994)
Nanoindentation: Localized Probes of Mechanical Behavior of Materials
16.84
16.85
16.86
16.87
16.88
16.89
16.90
16.91
16.92
16.94
16.95
16.96
16.97
16.98
16.99
16.100
16.101
16.102
16.103
16.104
16.105
16.106
16.107
16.108
16.109
W.W. Gerberich, N.I. Tymiak, J.C. Grunlan,
M.F. Horstemeyer, M.I. Baskes: Interpretations of
indentation size effects, J. Appl. Mech. 69, 433–442
(2002)
N. Gane, F.P. Bowden: Microdeformation of solids,
J. Appl. Phys. 39, 1432–1435 (1968)
J.B. Pethica, D. Tabor: Contact of characterized
metal surfaces at very low loads: Deformation and
adhesion, Surf. Sci. 89, 182–190 (1979)
T.A. Michalske, J.E. Houston: Dislocation nucleation
at nano-scale mechanical contacts, Acta Mater. 46,
391–396 (1998)
J.D. Kiely, J.E. Houston: Nanomechanical properties
of Au(111), (001), and (110) surfaces, Phys. Rev. B 57,
12588–12594 (1998)
D.F. Bahr, D.E. Wilson, D.A. Crowson: Energy considerations regarding yield points during indentation, J. Mater. Res. 14, 2269–2275
(1999)
A.H. Cotrell: Dislocations and Plastic Flow in Crystals (Oxford Press, Oxford 1953) pp. 11–12
A.M. Minor, S.A. Syed Asif, Z. Shan, E.A. Stach,
E. Cyrankowski, T.J. Wyrobek, O.L. Warren: A new
view of the onset of plasticity during the nanoindentation of aluminium, Nature Mater. 5, 697–702
(2006)
S.G. Corcoran, R.J. Colton, E.T. Lilleodden,
W.W. Gerberich: Anomalous plastic deformation at
surfaces: Nanoindentation of gold single crystals,
Phys. Rev. B 55, R16057–16060 (1997)
C.A. Schuh, J.K. Mason, A.C. Lund: Quantitative insight into dislocation nucleation from
high-temperature nanoindentation experiments,
Nature Mater. 4, 617–621 (2005)
A.H.W. Ngan, P.C. Wo: Delayed plasticity in nanoindentation of annealed crystals, Phil. Mag. 86,
1287–1304 (2006)
D.F. Bahr, G. Vasquez: The effects of solid solution impurities on dislocation nucleation during
nanoindentation, J. Mater. Res. 20, 1947–1951
(2005)
K.A. Nibur, D.F. Bahr, B.P. Somerday: Hydrogen effects on dislocation activity in austenitic stainless
steel, Acta Materialia 54, 2677–2684 (2006)
407
Part B 16
16.93
Q. Ma, D.R. Clarke: Size dependent hardness of
silver single crystals, J. Mater. Res. 10, 853–863
(1995)
A.A. Elmustafa, J.A. Eastman, M.N. Rittner,
J.R. Weertman, D.S. Stone: Indentation size effect:
large grained aluminum versus nanocrystalline
aluminum-zirconium alloys, Scripta Mater. 43,
951–955 (2000)
W.D. Nix, H. Gao: Indentation size effects in
crystalline materials: A law for strain gradient plasticity, J. Mech. Phys. Solids 46, 411–425 (1998)
S.J. Bull, T.F. Page, E.H. Yoffe: An explanation of
the indentation size effect in ceramics, Phil. Mag.
Lett. 59, 281–288 (1989)
M.D. Uchic, D.M. Dimiduk, J.N. Florando, W.D. Nix:
Sample dimensions influence strength and crystal
plasticity, Science 305, 986–989 (2004)
C.A. Volkert, E.T. Lilleodden: Size effects in the deformation of sub-micron Au columns, Phil. Mag.
86, 5567–5579 (2006)
C.A. Schuh, A.C. Lund: Application of nucleation
theory to the rate dependence of incipient plasticity during nanoindentation, J. Mater. Res. 19,
2152–2158 (2004)
S.A. Syed-Asif, J.B. Pethica: Nanoindentation creep
of single-crystal tungsten and gallium arsenide,
Philos. Mag. A 76, 1105–1118 (1997)
A.M. Minor, E.T. Lilleodden, E.A. Stach, J.W. Morris:
Direct observations of incipient plasticity during
nanoindentation of Al, J. Mater. Res. 19, 176–182
(2004)
D.F. Bahr, D.E. Kramer, W.W. Gerberich: Non-linear
deformation mechanisms during nanoindentation, Acta Mater. 46, 3605–3617 (1998)
A.B. Mann, J.B. Pethica: The role of atomic
size asperities in the mechanical deformation of
nanocontacts, Appl. Phys. Lett. 69, 907–909 (1996)
Y. Choi, K.J. V.Vliet, J. Li, S. Suresh: Size effects on
the onset of plastic deformation during nanoindentation of thin films and patterned lines, J. Appl.
Phys. 94, 6050–6058 (2003)
C.L. Kelchner, S.J. Plimpton, J.C. Hamilton: Dislocation nucleation and defect structure during surface
indentation, Phys. Rev. B 58, 11085–11088 (1998)
References