materials
Article
Effect of the Elastic Deformation of a Point-Sharp
Indenter on Nanoindentation Behavior
Takashi Akatsu 1,2, *, Shingo Numata 2 , Yutaka Shinoda 2 and Fumihiro Wakai 2
1
2
*
Faculty of Art and Regional Design, Saga University, 1 Honjo-machi, Saga 840-8502, Japan
Laboratory for Materials and Structures, Institute of Innovative Research, Tokyo Institute of Technology,
R3-24 4259 Nagatsuta, Midori, Yokohama 226-8503, Japan; [email protected] (S.N.);
[email protected] (Y.S.); [email protected] (F.W.)
Correspondence: [email protected]; Tel.: +81-952-28-8667
Academic Editors: Ting Tsui and Matt Pharr
Received: 25 January 2017; Accepted: 2 March 2017; Published: 7 March 2017
Abstract: The effect of the elastic deformation of a point-sharp indenter on the relationship between
the indentation load P and penetration depth h (P-h curve) is examined through the numerical
analysis of conical indentations simulated with the finite element method. The elastic deformation
appears as a decrease in the inclined face angle β, which is determined as a function of the elastic
modulus of the indenter, the parabolic coefficient of the P-h loading curve and relative residual depth,
regardless of h. This indicates that nominal indentations made using an elastic indenter are physically
equivalent to indentations made using a rigid indenter with the decreased β. The P-h curves for a
rigid indenter with the decreased β can be estimated from the nominal P-h curves obtained with
an elastic indenter by using a procedure proposed in this study. The elastic modulus, yield stress,
and indentation hardness can be correctly derived from the estimated P-h curves.
Keywords: nanoindentaion; elastic deformation; finite element method; numerical analysis
1. Introduction
Nanoindentation is a form of mechanical testing characterized as a depth-sensing indentation [1]
to evaluate local mechanical properties through the analysis of the indentation load P versus the
penetration depth h (P-h curve, hereafter). The analysis is principally based on a geometrical definition
in which the indentation is carried out on a flat surface using an indenter geometrically defined such
as flat-ended, spherical, ellipsoidal, point-sharp (e.g., conical, Berkovich, Vickers, cube corner, etc.).
The point-sharp indentation has an advantage in local mechanical testing owes to the analytical
simplicity for the geometrical similarity [2].
The bluntness of the indenter tip is one of the inevitable problems of undesirable tip geometry,
especially for the point-sharp indentations, because it is impossible to make an ideally sharp indenter.
The degree of the bluntness of a point-sharp indenter has been expressed in terms of the radius of
curvature at the tip [3–5], but the actual geometry of a blunt tip is not guaranteed to be spherical.
An area function [6,7] which gives the projected contact area at the maximum indentation load
is another approach to express the bluntness of a point-sharp indenter, but the area function is
theoretically valid only for hardness evaluation. A truncated tip which represents a blunt tip in an
extremely poor situation [8] is a suitable model for a strict discussion on the effect of the tip bluntness
on indentation behavior. According to the appendix of this paper, where a truncated tip is considered,
the undesirable effect of the bluntness of a point-sharp indenter can be removed out simply if the P-h
curve is shifted with ∆htip in the h direction for indentations deeper than 2∆htip , where ∆htip is the
distance between ideally sharp and blunt tips (see Figures A1–A4). In addition, ∆htip can be estimated
Materials 2017, 10, 270; doi:10.3390/ma10030270
www.mdpi.com/journal/materials
Materials 2017, 10, 270
2 of 12
through an extrapolation of the linear relationship between h and P*** observed in the large P and h
region to P = 0 (see Figure A5).
The elastic deformation of an actual point-sharp indenter, which has been conventionally taken
into account on the basis of Hertzian contact [6]; Hertzian contact was basically used for spherical
indentations as a modification of the elastic modulus evaluation. It is also an inevitable problem of
undesirable tip geometry, especially for indentations on a very hard material, and there is still some
controversy whether the modification based on the Hertzian contact can be applied to point-sharp
indentations. Moreover, there are no reports on the modifications of the indenter elastic deformation
for other mechanical properties such as the indentation hardness or yield stress. The geometrical
changes of a point-sharp indenter due to elastic deformation should be considered when evaluating
local mechanical properties with the nanoindentation technique.
In this paper, the effect of the elastic deformation of a point-sharp indenter on a P-h curve is
quantified in a numerical analysis of conical nanoindentation behaviors simulated with the finite
element method (FEM). In addition, a procedure of deriving physically meaningful P-h curves,
which should be utilized for mechanical property evaluation. Finally, the validity and accuracy
of this method is examined.
2. FEM Simulation of Nanoindentation
A conical indentation on a cylindrical elastoplastic solid was modeled in order to avoid the
difficulty of modeling a pyramidal indenter widely used for actual nanoindentations. The FEM
simulation exploited the large strain elastoplastic capability of ABAQUS code (Version 5.8.1) in the
same way as reported in the literature [9,10]. Indentation contact was simulated by the use of elastic
cone indenters with two different inclined face angles β (19.7◦ and 30◦ ). Young’s modulus of the elastic
indenter was in the range of 300–1140 GPa. The finite-element mesh in the elastic indenter with β of
19.7◦ was composed of 775 4-node quadrilateral axisymmetric elements with 2443 nodes. The elastic
indenter with β of 30◦ had 704 elements with 2258 nodes.
The FEM simulation used elastoplastic linear strain hardening rules, i.e., σ = Eε for σ < Y,
and σ = Y + E p ε p for σ ≥ Y, where σ is the stress, E the Young’s modulus and ε the strain. Here, Y is
the yield stress and Ep (≡ dσ/dε p ) is the plastic strain hardening modulus, where dσ, dε, dεe , and dεp
are, respectively, the incremental values of stress, total, elastic, and plastic strains. Indentations were
simulated for E, Y and Ep ranges of 50–1000 GPa, 0.1–60 GPa, and 0–200 GPa, respectively. The von
Mises criterion with isotropic hardening was used to determine the onset of yielding flow.
3. Results and Discussion
A quadratic relationship between P and h on loading is theoretically guaranteed for a point-sharp
indentation on the flat surface of a homogeneous elastoplastic solid [11,12]. The quadratic relationship
was also observed in simulated P-h curves made with an elastic cone indenter. This indicates that
the elastic deformation of a cone indenter can be described as a decrease in β determined regardless
of h. Therefore, nominal indentations made with an elastic cone indenter with an original inclined face
angle βo should be physically equivalent to indentations made with a rigid cone indenter with the
decreased inclined face angle βd .
A nominal quadratic P-h relationship for an elastic cone indenter can be depicted as follows:
P = k1n h2 for loading,
(1)
where k1n is the nominal indentation loading parameter. Here, h in Equation (1) is the nominal
penetration depth because the decrease in β from βo to βd , due to the elastic deformation of a cone
indenter, gives a decrease in real penetration depth. Thus, a physically meaningful P-h relationship can
be written with a true indentation loading parameter k1 , which should be observed in a P-h loading
curve using a rigid cone indenter with βd as
Materials 2017, 10, 270
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Pmax = k1 (hmax − ∆hd )2 ,
(2)
where ∆hd is the
decrease
in h at the maximum penetration depth hmax due to the elastic
deformation
Materials
2017, 10, 270
3 of 13
of a cone indenter (see Figure 1). The combination of Equations (1) and (2) leads to the equation:
Pmax  k1 hmax  hd −, 2
(2)
∆hd
k1 maximum
= k1n 1penetration
−
. max due to the elastic deformation
where Δhd is the decrease in h at the
hmaxdepth(1)hand
of a cone indenter (see Figure 1). The combination of Equations
(2) leads to the equation:
2
(3)
2
This means that ∆hd /hmax is a key parameter
k1 from the nominal k1n . In other
 to

hestimating
k1  k1n 1  d 
(3)
words, ∆hd /hmax can be simulated as
 hmax  .
s
This means that Δhd/hmax is a key parameter to estimating k1 from the nominal k1n. In other words,
∆h
k1n
d
Δhd/hmax can be simulated as
,
(3’)
= 1−
hmax
k1
hd
k
 1  1n
(3′)
hmax
k1 loading curve with an elastic cone indenter
where k1n in Equation (3’) is observed in a simulated
P-h
,
and k1 in Equation
(3’) is evaluated with the mechanical properties inputted into the FEM model [9,10].
where k1n in Equation (3’) is observed in a simulated P-h loading curve with an elastic cone indenter
k1 in Equation (3’)
evaluated
mechanical
inputted
the FEM model
[9,10].
In the followingand
paragraph,
theiseffect
ofwith
∆hthe
a P-h curve
is into
examined
quantitatively
through
max on properties
d /h
In the following paragraph, the effect of Δhd/hmax on a P-h curve is examined quantitatively through
numerical analysis.
numerical analysis.
Figure 1. Schematic
illustration
of the
effect
indenter
elastic
deformation
on a P-h curve.
Figure 1. Schematic
illustration
of the
effectof
of indenter
elastic
deformation
on a P-h curve.
In addition to k1, the relative residual penetration depth ξ, defined as hr/hmax, where hr is the
residual
depth, characterizes
a P-h curve anddepth
nominally
In addition
to k1penetration
, the relative
residual penetration
ξ, decreased
definedbyasthehrelastic
/hmax , where hr is
deformation of a cone indenter to be ξn. A true ξ-value, which should be observed in a P-h curve
the residual penetration
depth, characterizes a P-h curve and nominally decreased by the elastic
using a rigid cone indenter with βd, can also be evaluated with the mechanical properties inputted
deformation of into
a cone
indenter
to beThe
ξ nnumerical
. A trueanalysis
ξ-value,
which
be observed
in a P-h curve using
the FEM
model [9,10].
revealed
that should
the evaluated
ξ can be correlated
with
the
nominal
ξ
n as a function of Δhd/hmax
a rigid cone indenter with βd , can also be evaluated with the mechanical properties inputted into the
FEM model [9,10]. The numerical analysis revealed that the evaluated ξ can be correlated with the
nominal ξ n as a function of ∆hd /hmax
(
ξ = ξn 1 −
∆hd
hmax
0.85 )−0.50
.
(4)
Figure 2 plots ξ estimated with Equation (4) and ξ n against the true ξ evaluated with the
mechanical properties inputted into the FEM model [9,10]. The results indicate the validity of using
Equation (4) to estimate ξ from the nominal ξ n and ∆hd /hmax . In addition, it is confirmed that ξ n is
smaller than ξ because of the overestimation of the penetration depth h due to the elastic deformation
of the indenter. Moreover, a true indentation unloading parameter k2 defined as Pmax /(hmax − hr )*,

 hd 

 hmax 
   n 1  

Materials 2017, 10,



(4)
.
Figure 2 plots ξ estimated with Equation (4) and ξn against the true ξ evaluated with the
mechanical properties inputted into the FEM model [9,10]. The results indicate the validity of using
Equation (4) to estimate ξ from the nominal ξn and Δhd/hmax. In addition, it is confirmed that ξn is
270 smaller than ξ because of the overestimation of the penetration depth h due to the elastic deformation
of the indenter. Moreover, a true indentation unloading parameter k2 defined as Pmax/(hmax − hr)*,
which should be observed in an P-h unloading curve using a rigid cone indenter with βd, can be
estimated from a simulated P-h curve with an elastic cone indenter characterized by k1n and ξn using
(3) and
as P-h unloading curve using a rigid cone indenter
be Equations
observed
in(4)an
4 of 12
which should
with βd , can be
2
estimated from a simulated P-h curve with an elastic
cone
indenter
characterized
by
k

0
.
50
1n and ξ n using
2 

  h  0.85 
 hd  
k
d
1

Equations (3) and (4) as k 2 
 1   n 1  
 
 k1n 1 
(5)
2
1   

hmax  

 hmax 


(

 .

0.85 )−0.50 −2
−2
withdEquation (5) as well as the nominal
unloading
Figure
k1 3 plots the estimated k ∆h
∆hindentation
d

k2 =
=
k
1
−
1
−
ξ
1
−
n elastic cone indenter against the true
1nfrom a simulated P-h curve with an
parameter k determined
k
2
hmaxinputted into the FEM model [9,10].
hmax
ξ ) the mechanical properties
(1 −with
evaluated
Figure 3 indicates that
2
2n
2
.
(5)
k2 can be estimated correctly by using Equation (5) with Δhd/hmax, and that the nominal k2n is quite far
from k2 owing to the overestimation of h.
Figure 2. Relative residual depth ξ estimated with Equation (4) and ξn nominally observed plotted
Figure 2. Relativeagainst
residual
depth
ξ estimated
with
Equation
(4)model.
and ξ n nominally observed plotted
ξ evaluated
with mechanical
properties
inputted
into the FEM
against ξ evaluated
with mechanical properties inputted into the FEM model.
The numerical analysis also revealed that Δhd/hmax is determined to be
0.84


hd Equation
k
Figure 3Materials
plots2017,
the
k2 with
unloading
10, estimated
270
(6) 5 of 13
 0.616 (5) 1nas well
 as the nominal indentation

1
.
5
hmax P-h curve
 n an
parameter k2n determined from a simulated
cone
indenter
against
the true k2
 Ei 1 with
 elastic
,


where Ei’ is defined as Ei/(1 − νi*) and Ei and νi are Young’s modulus and Poisson’s ratio of an elastic
evaluated with
the mechanical
properties
into the FEM model [9,10]. Figure 3 indicates that
indenter,
respectively. Figure
4 plots Δhinputted
d/hmax estimated with Equation (6) against Δhd/hmax evaluated
k2 can be estimated
correctly
by using
Equation
and that
the(6)
nominal
with Equation
(3’). Figure
3 indicates
that Δhd/h(5)
max with
can be ∆h
estimated
by, using
Equation
with a k2n is quite
d /hmax
nominally
P-h curve characterized
far from k2 owing
to observed
the overestimation
of h. by k1n and ξn, and with the elastic properties of an elastic
indenter characterized by Ei and νi.
Figure 3. Indentation unloading parameter k2 simulated and k2n nominally observed plotted against
Figure 3. Indentation
unloading
parameter
k2 simulated
andmodel.
k2n nominally observed plotted against
mechanical
properties inputted
into the FEM
k2 evaluated with
k2 evaluated with mechanical properties inputted into the FEM model.
The numerical analysis also revealed that ∆hd /hmax is determined to be
0.84



∆hd
k
1n
= 0.616
E 0 1 + ξ
hmax
i
n
1.5
,
(6)

where Ei 0 is defined as Ei /(1 − νi *) and Ei and νi are Young’s modulus and Poisson’s ratio of an
elastic indenter, respectively. Figure 4 plots ∆hd /hmax estimated with Equation (6) against ∆hd /hmax
Figure 4. Degree of elastic deformation of conical indenter Δhd/hmax estimated with Equation (6)
plotted against Δhd/hmax evaluated with Equation (3’).
Materials 2017, 10, 270
5 of 12
evaluated with Equation (3’). Figure 3 indicates that ∆hd /hmax can be estimated by using Equation (6)
with a nominally observed P-h curve characterized by k1n and ξ n , and with the elastic properties of an
Figure 3. Indentation
parameter
k simulated and k nominally observed plotted against
elastic indenter characterized
by Eunloading
i and ν
i.
2
2n
k2 evaluated with mechanical properties inputted into the FEM model.
Figure 4. Degree of elastic deformation of conical indenter Δhd/hmax estimated with Equation (6)
against Δh
d/hmax evaluated with
(3’).indenter ∆hd /hmax estimated with Equation (6)
Figure 4. Degree plotted
of elastic
deformation
ofEquation
conical
plotted against ∆hInd /h
evaluated
with
Equation
(3’).
max
order
to estimate mechanical properties from
a P-h curve characterized with k1, k2 and ξ, we
should know the inclined face angle βd of the elastically deformed indenter. Numerical analysis
revealed that βd is given as a function of βo, ξn and Δhd/hmax
In order to estimate mechanical properties from a P-h 0curve
characterized with k1 , k2 and ξ,
.90
tan βd of the elastically
0.80 0.83  hd 
we should know the inclined face angle
deformed
indenter. (7)
Numerical analysis



1

1


d
n
h 
max 
 max
revealed that βd is given as a functiontan
ofβo o , ξ n and ∆hd /h
.

tan 

tan 
d
d
estimated with Equation (7) against
observed in a simulated
0.83 ∆h
tan 
tan  o 0.90
tan
βod
d
0.80
=
1
−
1
−
ξ
n
nanoindentation, and indicates the validity to estimate
the inclined face angle β. of the elastically
tan β
hmax
deformed indenter with Δho /h .
Figure 5 plots
d
d
(7)
max
tan β
tan β
Figure 5 plots tan βdo estimated with Equation (7) against tan βdo observed in a simulated
nanoindentation, and indicates the validity to estimate the inclined face angle βd of the elastically
deformed indenter with ∆hd /hmax .
Materials 2017, 10, 270
6 of 13
Figure 5. Degree of change in inclined face angle tanβd/tanβo estimated with Equation (7) plotted
o observed in simulated nanoindentation.
tanβd/tanβin
Figure 5. Degree against
of change
inclined face angle tanβd /tanβo estimated with Equation (7) plotted
against tanβd /tanβo observed in simulated nanoindentation.
E
The representative indentation elastic modulus E*, defined as *
E 

1    0.225 tan 1.05  d

2
in terms of βd [9], can be estimated from the simulated P-h curve using Equations (3)–(7) if we know
E
2 in terms
i
i
d observed in
1−
ν−0.225 tan1.05 β d )
(
simulated nanoindentations. Figure 6 plots the estimated E* (black circles) against the evaluated E*.
of βd [9], can be estimated
from
the
simulated P-h curve using Equations (3)–(7) if we know Ei and νi ,
The white circles
are E*
n estimated with the nominal values of k2n and ξn [9], which means that the
elastic deformation
of an indenter
is not modified
for FEM
the estimation
of E*.
Figure
6 indicates
the
whereas it is evaluated
with E and
ν inputted
into the
model
and
with
βd observed
in simulated
modification of the elastic deformation of an indenter can determine E* correctly. The
nanoindentations.underestimation
Figure 6 plots
the
estimated
E*
(black
circles)
against
the
evaluated
E*.
The white
of E* without the modification (the white circles in Figure 6) is caused by the
overestimation
of
the
elastic
rebound
during
the
unloading
process
because
the
extrinsic
elastic
circles are E*n estimated with the nominal values of k2n and ξ n [9], which means that the elastic
deformation of the indenter is added to the intrinsic elastic deformation of the indented material.
The representative
indentation elastic modulus E*, defined as E∗ =
E and ν , whereas it is evaluated with E and ν inputted into the FEM model and with β
deformation of an indenter is not modified for the estimation of E*. Figure 6 indicates the modification
of the elastic deformation of an indenter can determine E* correctly. The underestimation of E* without
the modification (the white circles in Figure 6) is caused by the overestimation of the elastic rebound
against tanβd/tanβo observed in simulated nanoindentation.
The representative indentation elastic modulus E*, defined as E * 
E

1    0.225 tan 1.05  d

2
in terms of βd [9], can be estimated from the simulated P-h curve using Equations (3)–(7) if we know
Ei and νi, whereas it is evaluated with E and ν inputted into the FEM model and with βd observed in
Materials 2017, 10, 270simulated nanoindentations. Figure 6 plots the estimated E* (black circles) against the evaluated E*.
The white circles are E*n estimated with the nominal values of k2n and ξn [9], which means that the
elastic deformation of an indenter is not modified for the estimation of E*. Figure 6 indicates the
modification of the elastic deformation of an indenter can determine E* correctly. The
underestimation
E* without the
the modification
white deformation
circles in Figure 6)of
is the
caused
by the
during the unloading
processofbecause
extrinsic (the
elastic
indenter
overestimation of the elastic rebound during the unloading process because the extrinsic elastic
intrinsic elastic deformation
of
the
indented
material.
deformation of the indenter is added to the intrinsic elastic deformation of the indented material.
6 of 12
is added to the
Figure 6. Representative indentation elastic modulus E* estimated with simulated P-h curves plotted
against E* evaluated
with mechanical
properties
inputted
the FEM model.
Figure 6. Representative
indentation
elastic
modulus
E*into
estimated
with simulated P-h curves plotted
against E* evaluated with mechanical properties inputted into the FEM
Y  0model.
.25E tan 
*
The representative indentation yield stress Y*, defined as Y 
The

p
d
in

1    0.225 tan 1.05  d
Y +0.25Ep tan β d
terms of βd [10], can also be estimated using a simulated P-h curve and Equations
representative
indentation yield stress Y*, defined as Y ∗ = (3)–(7) if we know
1−(ν−0.225 tan1.05 β d )
in terms of
βd [10], can also be estimated using a simulated P-h curve and Equations (3)–(7) if we know Ei and
νi . Moreover, it can be evaluated with Y, Ep and ν inputted into the FEM model and with βd of the
simulated indentation. Figure 7 plots the estimated Y* (black circles) against the evaluated Y*. Y*n
estimated with E*Materials
and
10, 270βo is plotted for comparison. This figure shows that
7 of 13 the modification
n , ξ n2017,
of the elastic deformation
of
an
indenter
more
or
less
correctly
estimates
Y*
although
the difference
E and ν . Moreover, it can be evaluated with Y, E and ν inputted into the FEM model and with β of
the simulated
Figure 7Y*
plotsisthe
estimated
Y* (black
circles)respect
against theto
evaluated
Y*. Y*
between the modified
andindentation.
unmodified
not
so large
with
the difference
observed in
estimated with E* , ξ and β is plotted for comparison. This figure shows that the modification of the
E* (see Figure 6). elastic
A relatively
difference
incorrectly
Y* isestimates
typically
found
in the range
deformation large
of an indenter
more or less
Y* although
the difference
between of ξ less than 0.1,
the modified and
unmodified
Y* is not The
so large
with respect
to the difference
observed
E* (see 7 is attributed to
where plastic deformation
is not
dominant.
small
difference
observed
in inFigure
Figure 6). A relatively large difference in Y* is typically found in the range of ξ less than 0.1, where
the decrease in k1nplastic
anddeformation
ξ n due to
elastic
deformation
of anobserved
elasticin indenter,
wheretothe
is not
dominant.
The small difference
Figure 7 is attributed
the former decreases
decrease
in k andincreases
ξ due to elastic
of an elastic indenter, where the former decreases Y*
Y* nominally while
the latter
Y*deformation
apparently.
i
i
p
d
n
n
1n
n
o
n
nominally while the latter increases Y* apparently.
Figure 7. Representative indentation yield stress Y* estimated with simulated P-h curves plotted
against Y* evaluated
with mechanical
properties
into the FEM model.
Figure 7. Representative
indentation
yield
stressinputted
Y* estimated
with simulated P-h curves plotted
against Y* evaluated
with
mechanical
properties
inputted
into
the
A previous study on the indentation hardness HM found that it FEM
can bemodel.
evaluated with the
mechanical properties inputted into the FEM model and with the simulated βd [9,10]. On the other
hand, HM can be estimated from a true P-h curve characterized with k1, k2 and ξ. Figure 8 plots the
estimated on
HM (black
against the evaluated
HM. The
HMn estimated
the nominal
A previous study
the circles)
indentation
hardness
Hnominal
that itwith
can
be evaluated with the
M found
P-h curve is plotted as white circles in Figure 8, and a comparison reveals that the modification more
mechanical properties
inputted
into
the
FEM
model
and
with
the
simulated
βd [9,10]. On the other
or less correctly estimates HM, although the difference between the estimated HM and the nominal
Mn is not so large
with respect
to the
difference
in E* (see Figure
6). Thekdifference
is
rather
hand, HM can be Hestimated
from
a true
P-h
curveobserved
characterized
with
,
k
and
ξ. Figure 8 plots the
1 2
high in the large HM region, where elastic deformation of the indenter is most severe. The small
estimated HM (black
circles) against the evaluated H . The nominal H deformation
estimated
with the nominal
difference observed in Figure 8 owes to the decreaseMin k1n and ξn due to elasticMn
of an
elastic
the former
decreases
M nominally
while the latterreveals
increases H
M apparently
P-h curve is plotted
asindenter,
white where
circles
in Figure
8,Hand
a comparison
that
the modification more
through the decrease of nominal contact depth.
or less correctly estimates
HMnanoindentation
, althoughexperiments
the difference
between
the
estimated
HM and the nominal
We conducted
and reported
E*, Y* and HM
for several
materials [9,10].
These
values
were
evaluated
with
modified
P-h
curves
(see
Figure
9)
considering
elastic
deformation
HMn is not so large with respect to the difference observed in E* (see Figure 6). The difference is rather
of a diamond indenter with Young’s modulus and a Poisson’s ratio of 1140 GPa and 0.07, respectively.
high in the large Table
HM 1region,
elastic
deformation
of the
indenter
iscurve
most
severe. The small
shows thesewhere
mechanical
properties
as well as those evaluated
with
a nominal P-h
made
without considering any elastic deformation of the indenter. Δhd/hmax and βd estimated with the
numerical analysis developed in this study are shown in order to examine the degree of the elastic
deformation of the indenter. Δhd/hmax and the change in β (Δβ = βo − βd) are large for relatively hard
materials (e.g., fused silica and alumina), which would cause a large elastic deformation of the
indenter. Even in that case, the changes in Y* and HM due to the elastic deformation of the indenter
are not so large. In contrast, the change in E* is so large that it cannot be ignored. The underestimation
of E* without the modification is caused by the overestimation of the elastic deformation during the
unloading process because the extrinsic elastic deformation of the indenter is added to the intrinsic
Materials 2017, 10, 270
7 of 12
difference observed in Figure 8 owes to the decrease in k1n and ξ n due to elastic deformation of an
elastic indenter, where the former decreases HM nominally while the latter increases HM apparently
through the decrease of nominal contact depth.
Materials 2017, 10, 270
9 of 13
Materials 2017, 10, 270
9 of 13
Figure 8. Indentation hardness HM estimated with simulated P-h curves plotted against HM evaluated
with mechanical properties inputted into the FEM model and with simulated βd.
Figure 8. Indentation
hardness HM estimated with simulated P-h curves plotted against HM evaluated
with mechanical properties inputted into the FEM model and with simulated βd .
We conducted nanoindentation experiments and reported E*, Y* and HM for several materials [9,10].
These values were evaluated with modified P-h curves (see Figure 9) considering elastic deformation
of a diamond indenter with Young’s modulus and a Poisson’s ratio of 1140 GPa and 0.07, respectively.
Table 1 shows these mechanical properties as well as those evaluated with a nominal P-h curve made
without considering any elastic deformation of the indenter. ∆hd /hmax and βd estimated with the
numerical analysis developed in this study are shown in order to examine the degree of the elastic
deformation of the indenter. ∆hd /hmax and the change in β (∆β = βo − βd ) are large for relatively
hard materials (e.g., fused silica and alumina), which would cause a large elastic deformation of the
indenter. Even in that case, the changes in Y* and HM due to the elastic deformation of the indenter
Figure 9. Flow chart of the procedure to evaluate mechanical properties.
are not so large. In contrast, the
change in E* is so large that it cannot be ignored. The underestimation
According to Equation
(6), the following
can be derived of the elastic deformation during the
of E* without the modification
is caused
by theequation
overestimation
1
unloading
process
becausehardness
the extrinsic
deformation
the indenter
is added
to the intrinsic
0.84
2
Figure
8. Indentation
HHM estimated
with
simulated
curves
plotted against
HM evaluated
 of
hd P-h
elastic
1 .5 
M


(8)



1


n
d.
with mechanical
inputted
into
the
FEM
model
and
with
simulated
β


elastic deformation
of properties
the indented
material.
E'
g
0.616h

i
max

.
When indentation hardness is not affected much by the indenter elastic deformation, where γ is
the surface profile parameter defined as γ = hmax/hc, hc is the contact depth, and g is the geometrical
factor of a point-sharp indenter to be 24.5 for β = 19.7°. Ei’ is required to be about 250 times larger than
HM for Δhd/hmax smaller than 0.05, where the effect of the indenter elastic deformation on a P-h curve
may be ignored for indentations with Berkovich-type indenter.
Figure 9. Flow chart of the procedure to evaluate mechanical properties.
Figure 9. Flow chart of the procedure to evaluate mechanical properties.
According to Equation (6), the following equation can be derived
HM  2
1 .5

1 n
Ei '
g


 hd

 0.616hmax



1
0.84
(8)
.
Materials 2017, 10, 270
8 of 12
Table 1. Effect of elastic deformation of diamond indenter on mechanical property evaluations.
Materials
∆hd /hmax
βd /βo (deg.)
k2 /k2n (10 3 GPa)
ξ/ξ n
E* (GPa)
Y* (MPa)
H M (GPa)
(Modified Value/Unmodified Value)
Brass
Duralumin
Beryllium copper alloy
Fused silica
Alumina
0.020
0.020
0.027
0.065
0.158
19.6/19.7
19.6/19.7
19.6/19.7
19.0/19.7
18.3/19.7
7.65/4.78
4.37/3.03
9.49/5.44
0.615/0.477
4.90/2.25
0.930/0.913
0.909/0.893
0.925/0.904
0.550/0.522
0.690/0.614
102/81
77/64
136/103
74/65
340/225
584/597
605/618
841/863
5.21 × 103 /5.35 × 103
12.6 × 103 /12.9 × 103
1.26/1.29
1.31/1.33
1.82/1.86
8.56/8.40
24.3/22.8
Materials 2017, 10, 270
9 of 12
According to Equation (6), the following equation can be derived
HM
γ2 1.5
1
+
ξ
=
n
g
Ei 0
∆hd
0.616hmax
1/0.84
(8)
When indentation hardness is not affected much by the indenter elastic deformation, where γ is
the surface profile parameter defined as γ = hmax /hc , hc is the contact depth, and g is the geometrical
factor of a point-sharp indenter to be 24.5 for β = 19.7◦ . Ei ’ is required to be about 250 times larger than
HM for ∆hd /hmax smaller than 0.05, where the effect of the indenter elastic deformation on a P-h curve
Materials
2017, 10,for
270 indentations with Berkovich-type indenter.
10 of 13
may
be ignored
4. Conclusions
4. Conclusions
The
effect
geometrical
changes
due
elastic
deformation
a point-sharp
indenter
The
effect
of of
thethe
geometrical
changes
due
to to
thethe
elastic
deformation
of of
a point-sharp
indenter
was
was
examined
by
conducting
a
numerical
analysis
of
P-h
curves
simulated
with
FEM.
The
effect
examined by conducting a numerical analysis of P-h curves simulated with FEM. The effect appears as
appears as a decrease in the inclined face angle β. The key parameter Δhd/hmax, which can be utilized
a decrease in the inclined face angle β. The key parameter ∆hd /hmax , which can be utilized to derive
to derive the physically meaningful P-h curve and the decreased β, can be estimated with an equation
the physically meaningful P-h curve and the decreased β, can be estimated with an equation derived
derived by numerical analysis. The mechanical properties of indented materials, such as E*, Y* and
by numerical analysis. The mechanical properties of indented materials, such as E*, Y* and HM , can be
HM, can be estimated by using the P-h curve and β characterized by k1, k2 and ξ estimated with the
estimated by using the P-h curve and β characterized by k1 , k2 and ξ estimated with the key parameter
key parameter Δhd/hmax. The modification of a P-h curve and β with Δhd/hmax is most effective for the
∆hd /hmax . The modification of a P-h curve and β with ∆hd /hmax is most effective for the estimation of
estimation of the accurate E* with respect to Y* and HM.
the accurate E* with respect to Y* and HM .
Acknowledgments: Financial support for this research was provided in the form of a Grant-In-Aid for Scientific
support
for this research
was provided
in the form
of athe
Grant-In-Aid
for Scientific
Acknowledgments:
Research (C) (No.Financial
18560652),
Encouragement
of Scientists
(No. 13750622)
and
Nippon Sheet
Glass
Research (C) (No. 18560652), Encouragement of Scientists (No. 13750622) and the Nippon Sheet Glass Foundation
Foundation for Materials Science and Engineering.
for Materials Science and Engineering.
Author
Contributions:
TakashiAkatsu
Akatsuconceived
conceived of
of the
the ideas
ideas for
analysis
Takashi
for this
this research,
research,performed
performednumerical
numerical
analysis
Author
Contributions:
simulated
nanoindentationsand
and wrote
wrote the
the paper.
paper. Shingo
on on
thethe
simulated
nanoindentations
Shingo Numata
Numataand
andYutaka
YutakaShinoda
Shinodasimulated
simulated
nanoindentation.
FumihiroWakai
Wakaiadvised
advisedon
onthe
the numerical
numerical analysis
of of
thethe
paper.
nanoindentation.
Fumihiro
analysisand
andreviewed
reviewedthe
themanuscript
manuscript
paper.
Conflicts
of of
Interest:
The
authors
Conflicts
Interest:
The
authorsdeclare
declareno
noconflict
conflictof
ofinterest.
interest.
Appendix.
of the
TipTip
Bluntness
of of
a Point-Sharp
AppendixEffect
A. Effect
of the
Bluntness
a Point-SharpIndenter
Indenteron
onNanoindentation Behavior
Nanoindentation Behavior
An indentation on a linearly elastic solid with a Young’s modulus E and Poisson’s ratio ν
An indentation
a linearly elastic
solid withusing
a Young’s
modulus
E and
Poisson’s
ratio tip
ν of[8]
of 100 GPa
and 0.499, on
respectively,
was simulated
a rigid
indenter
with
a truncated
100 GPa
0.499, respectively,
was simulated
using a rigid
indenter
with was
a truncated
tipcarried
[8] in order
in order
to and
represent
a tip in an extremely
poor situation.
The
simulation
basically
out in
to
represent
a
tip
in
an
extremely
poor
situation.
The
simulation
was
basically
carried
out
in
the
same
the same way described in the Section 2 of this paper. The inclined face angle β of the indenter was
in theisSection
2 of this paper. The
inclined face
angle
of the indenter
was set to
be
setway
to bedescribed
19.7◦ , which
the Vickers–Berkovich
equivalent
angle.
Theβdistance
∆htip between
ideally
19.7°,
which
is the Vickers–Berkovich
equivalent
Theor
distance
Δhwith
tip between ideally sharp and
sharp
and
truncated
tips (see Figure A1)
was set angle.
to be 716
1790 nm
respect to the maximum
truncated
tips
(see
Figure
A1)
was
set
to
be
716
or
1790
nm
with
respect
to
the
maximum penetration
penetration depth of 10 µm.
depth of 10 μm.
Figure A1. Schematic illustration of a truncated conical indenter tip.
Figure A1. Schematic illustration of a truncated conical indenter tip.
The P-h curves shown in Figure A2 were obtained by simulating an indentation on a perfectly
The P-h
in Figure
by simulating
indentation
a perfectly
 0.5obtained
elastic
bodycurves
with ashown
Poisson’s
ratio νA2
of were
for a series
of Δhtip. Weanonly
examinedon
a perfectly
elastic
body
with
a Poisson’s
ratioofνbluntness
of ≈ 0.5 for
a series
of ∆hsevere
.
We
only
examined
a
perfectly
elastic
elastic
body
because
the effect
would
be most
in
this
case.
As
shown
in
Figure
tip
A3, the curve deviates from that for an ideally sharp indenter (Δhtip = 0) as Δhtip increases and the
amount of the shift in the h direction almost coincides with Δhtip. The results in this figure indicate
the effect of a blunt tip of a conical indenter on the P-h curve can be simply described as a change in
the penetration depth with Δhtip, especially in the large P and h region. The error in Young’s modulus
E P  h  Einp
Materials 2017, 10, 270
10 of 12
body because the effect of bluntness would be most severe in this case. As shown in Figure A3, the
curve deviates from that for an ideally sharp indenter (∆htip = 0) as ∆htip increases and the amount of
the shift in the h direction almost coincides with ∆htip . The results in this figure indicate the effect of a
blunt tip of a conical indenter on the P-h curve can be simply described as a change in the penetration
depth with ∆htip , especially in the large P and h region. The error in Young’s modulus is evaluated
|E −E |
as P−Ehinp inp (expressed as a percentage), where EP-h and Einp are Young’s modulus values obtained
from the simulated P-h curve and inputted into an FEM model, respectively. Figure A4 plots this error
as a function of penetration depth normalized by ∆htip . It indicates that a reliable Young’ modulus
can be obtained with an error less than 10% from the P-h curve in the h region larger than 2∆htip .
In other words, P-h data for h-values shallower than 2∆htip should be omitted in order to obtain reliable
mechanical properties. It may be possible to get more reliable P-h data in the shallow h region for
other geometries,
but the P-h data are actually obscure and depend on the unknown
geometry of
Materials 2017, 10, 270
11 of 13
the indenter tip. We simulated a truncated tip of a conical indenter because it represents the worst
Materials
2017,
10, other
270
11 of to
13
than 2Δh
tip. In
words, P-h data for h-values shallower than 2Δhtip should be omitted in order
case of a blunt obtain
point-sharp
indenter;
consequently,
thetoP-h
datareliable
obtained
h region larger than
reliable mechanical
properties.
It may be possible
get more
P-h datain
in the
the shallow
than
2Δhtip
. Inother
othergeometries,
P-h but
datathe
forP-h
h-values
shallower
than
tip should be omitted in order to
h region
for
data
are
actually
and depend
the unknown
2∆htip guarantees
accuracy
inwords,
all cases.
Figure
A5,
where
hobscure
is2Δh
plotted
as aon
function
of the square root
obtain
reliable
properties.
It may
be possibletip
toofget
more reliable
P-hbecause
data initthe
shallow
geometry
of themechanical
indenter tip.
We simulated
a truncated
a conical
indenter
represents
of P, is an extrapolation
of
the
linear
relationship
between
h
and
P***
observed
in
the large P and h
hthe
region
other
but theindenter;
P-h data consequently,
are actually obscure
oninthe
worstfor
case
of ageometries,
blunt point-sharp
the P-hand
datadepend
obtained
theunknown
h region
geometry
of
the
indenter
tip.
We
simulated
a
truncated
tip
of
a
conical
indenter
because
it
represents
region to P = 0.larger
It gives
∆h
as an accuracy
absolute
on the
and
is aas way
to estimate
∆htip from
than 2Δh
tip tip
guarantees
in allvalue
cases. Figure
A5, h-axis
where h is
plotted
a function
of the
the
worst
blunt
point-sharp
indenter;
consequently,
the P-hh data
obtained
in the
h region
square
rootcase
of P,ofisaan
extrapolation
of the
linear relationship
between
and P***
observed
in the
large
the P-h curve. larger
We have
already
reported
an
example
of
estimating
∆h
from
an
experimental
P-h
tip
2Δhto
tip guarantees accuracy in all cases. Figure A5, where h is plotted as a function of the
P and hthan
region
P = 0. It gives Δhtip as an absolute value on the h-axis and is a way to estimate Δhtip
square
root
of
P,
is
an
extrapolation
of
the
linear
relationship
between
h
and
P***
observed
in
the
large
curve [13]; ∆htip
is
estimated
to
be
about
50
nm
for
the
Berkovich
indenter
that
we
actually
used
for
from the P-h curve. We have already reported an example of estimating Δhtip from an experimental
P
and
h region
to
P
= 0. It gives Δhtip as an absolute value on the h-axis and is a way to estimate Δhtip
P-h
curve
[13];
Δh
tip is estimated to be about 50 nm for the Berkovich indenter that we actually used
the nanoindentation
experiment.
from the P-h curve. We have already reported an example of estimating Δhtip from an experimental
for the nanoindentation experiment.
P-h curve [13]; Δhtip is estimated to be about 50 nm for the Berkovich indenter that we actually used
for the nanoindentation experiment.
Figure A2. Indentation load P vs. penetration depth h curves obtained with truncated conical indenters.
Figure A2. Indentation load P vs. penetration depth h curves obtained with truncated conical indenters.
Figure A2. Indentation load P vs. penetration depth h curves obtained with truncated conical indenters.
Figure A3. Indentation load P vs. modified penetration depth h + Δhtip curves obtained with truncated
conical indenters.
Figure A3. Indentation load P vs. modified penetration depth h + Δhtip curves obtained with truncated
Indentation
load P vs. modified penetration depth h + ∆htip curves obtained
conical indenters.
Figure A3.
conical indenters.
with truncated
Materials 2017, 10, 270
11 of 12
Materials 2017, 10, 270
12 of 13
Materials 2017, 10, 270
12 of 13
Figure A4. Error in Young’s modulus evaluation with a truncated conical indenter as a function of
Figure A4. Errornormalized
in Young’s
modulus
with a truncated conical indenter as a function of
penetration
depth (h +evaluation
Δhtip)/Δhtip.
normalized penetration
depth
(h + ∆h
Figure A4. Error
in Young’s
modulus
evaluation
with a truncated conical indenter as a function of
tip )/∆h
tip .
normalized penetration depth (h + Δhtip)/Δhtip.
Figure A5. Penetration depth h vs. square root of indentation load P*** obtained with truncated indenters.
References
1.
Figure A5. Penetration depth h vs. square root of indentation load P*** obtained with truncated indenters.
Mann, A.B.; Cammarata, R.C.; Nastasi, M.A. Nanoindentation: From angstroms to microns—Introduction.
Figure A5. Penetration
depth h vs. square root of indentation load P*** obtained with truncated indenters.
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