SPHERICAL AND ROUNDED CONE NANO INDENTERS
Bernard Mesa
Micro Star Technologies
In the present field of nano indentation, spherical tipped indenters made of diamond or sapphire are
desirable in numerous applications. A truly spherical tipped cone, as in Fig. 1, is difficult to fabricate at
nanometer scale. In practice, a rounded cone may have a geometry similar to Fig. 2. The tip is spherical
at the apex but has a transition section which is neither part of the sphere nor the cone. If only a
minimal indentation depth is sufficient, such a rounded cone provides acceptable spherical indentations.
When deeper indentations are needed, a more precise definition of the area function is required.
Figure 1. Spherical tipped cone profile.
Figure 2. Rounded tipped cone profile.
The analysis in the following pages offers a means to calculate the area function of rounded tip
indenters with a single equation that is valid for both perfectly spherical and rounded cones.
First, the area function equations for the sphere, the cone and the spherical tipped cone are provided.
Then the rounded cone equation and its application are described. The calculated area function values
at regular indenting intervals are given in a spread sheet table.
Appendix A shows the equation derivation and Appendix B provides actual examples of rounded cone
indenters analysis.
MST manufactures diamond and sapphire cone nano indenters with rounded tips at micrometer and
nanometer dimensions. A TEM calibrated with a traceable standard is used to image and measure most
of its nano indenters.
The graphic and calculated analysis of rounded conical indenters described here is available on request
for purchased indenters. When ordering rounded cone indenters please supply the expected depth of
indentation, in addition to the desired tip radius and cone angle.
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THEORETICAL SPHERE AND CONE AREA FUNCTIONS
Figure 2. Spherical tip cone
A cone indenter with a perfect spherical tip is shown on Fig. 2. The nomenclature used is as follows.
R
h
r
α
T
C
P
O
a
Sphere radius
Indentation depth
Radius of projected circle at indentation depth
Cone half angle
Transition between cone and sphere
Sphere center
Indenter apex
Cone theoretical apex
Distance from P to O
An indenter area function f(h) allows the calculation of the projected area A of the circle of radius r at
indentation depth h. Equation (2) is valid for all conical indenters which are assumed to have a circular
symmetry.
r = f(h)
A=π r
(1)
2
(2)
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Figure 3. Spherical Indenter
Figure 4. Cone indenter
Simple spherical indenter equations,
r2 = R2 – (R‐h)2
(3)
r2 = 2Rh – h2
(4)
A = π (2Rh – h2)
(5)
Simple conical indenter equations,
r = h tan α
(6)
A = π h2 tan 2 α
(7)
Figure 5. Spherical tip cone
hT
Indentation depth at the transition T between sphere and cone
rT
Radius of projected circle at transition depth
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Equations for the spherical section, when h ≤ hT:
r2 = 2Rh – h2
(4)
A = π (2Rh – h2)
(5)
At the transition, when h = hT :
Sin α = (R – hT ) / R
(8)
hT = R (1 – Sin α)
(9)
rT = R Cos α
(10)
Equations for the conical section, when h ≥ hT:
Tan α = r / (a + h)
(11)
r = Tan α (a + h)
(12)
A = π [Tan α (a + h)]2
(13)
At the transition, when h = hT :
rT = Tan α ( a + hT )
(12)
Sin α = R / (R + a)
(14)
a = R ( 1 / Sin α – 1 )
(15)
hT = R ( 1 – Sin α )
(9)
rT = Tan α [R ( 1 / Sin α – 1 ) + ( 1 – Sin α )]
(16)
rT = R Tan α ( 1 / Sin α – Sin α )
(17)
rT = R ( 1 / Cos α – Sin2 α / Cos α )
(18)
rT = R [ 1– (1 – Cos2 α)] / Cos α
(19)
rT = R Cos α
(20)
Which is the same result for rT from the sphere:
rT = R Cos α
(10)
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ACTUAL ROUNDED CONE NANO INDENTERS
Actual diamond nano indenters that approach a perfect spherical tip can only be made with
considerable extra time and effort. There are two main reasons. One is the anisotropy of diamond
which offers different abrasion rates at different crystal directions. This hampers circular symmetry.
The second reason is the very small dimensions required. At micro and nano meter scales the processes
are not precise and repeatable enough to directly produce the desired geometries. These can only be
approached by repeating the process in many small steps followed by measurements (usually with an
electron microscope) until the required dimensions and tolerances are achieved.
Figs. 6, 7 and 8 show transmission electron microscope (TEM) images of three indenter examples. On
the left is the plain TEM image. On the right some graphics have been superimposed. The larger circle
indicates the sphere that would fit tangent to the cone sides. An spherical surface in this position would
make the ideal spherical indenter.
The smaller circle is a closer approximation to the curve at the indenter tip. If the indentation depths
are small in relation to the circle (less than 20% of the small circle radius), the indenter is acceptable as
spherical. At deeper indentations the small circle radius would not be a good basis for accurate
measurements.
Figure 6. TEM image of indenter VR13211
Figure 7. TEM image of indenter VR13212
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Figure 8. TEM image of indenter VR13240
An investigation has been done on the non spherical geometry indenters to determine their area
function general equation. There are two equations that provide the projected area as a function of the
indentation depth. Equation (21) is applicable to the rounded section of the indenter and equation (12)
to the conical section. Appendix A describes in detail the derivation of equation (21).
Radius of the projected circle at an indentation depth h, when h ≤ hT:
r2 = 2(RP + (RT ‐ RP) KhK / hT)h ‐ h2
(21)
Radius of the projected circle at an indentation depth h, when h ≤ hT:
r = Tan α (a + h)
(12)
In both cases,
A = π r2
(2)
Fig. 9 shows the TEM image of indenter VR13211 with the measurement parameters required by
equation (21). The two lines TO and T’O are the cone sides meeting at O. T is the transition where the
tip’s curve starts. At point T a perpendicular line extended to the indenters center is the large circle
radius or RT. The small circle radius RP is determined at a point where h is 2.5% of hT as explained on the
Appendix. Following is the nomenclature for equation (21) and Fig. 9 not defined on page 2.
RP
RT
hT
rT
K
Apex circle radius
Transition point circle radius
Indentation depth at transition
Projected radius at transition depth
Adjustable h coefficient and exponent.
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Figure 9. TEM image with measuring parameters.
MST provides, on request, the analysis of a particular rounded cone indenter. For this purpose, the
indenter’s TEM image is measured on a CAD program set to the microscope scale at which the image
was taken. Fig. 10 shows the graphic analysis of indenter VR 13211 as an example.
Table 1 is the spread sheet where the parameters have been entered. Equations (21 ) and (12) are used
to calculate a series of values for r and A at equally spaced h intervals. Notice that rT (at h = hT = 2.200)
is calculated independently with equations (21) and (12). The results differ slightly because the 3
significant decimal precision may round the values in some of the calculations.
The “K factor” is a number used to adjust equation (21). K values fall between 1.00 and 0.70. The value
of K is adjusted empirically to minimize the difference between rT calculated and rT measured. On
Table 1 rT calculated with equation (21) is 2.047, rT measured is 2.049 using K = 0.890.
In the appendix several different indenters are measured point by point and compared to the calculated
values, showing the validity of equation (21). In the case of a perfect spherical indenter RT = RP = R and,
equation (21) becomes equation (4),
r2 = 2(RP + (RT – RP)KhK / hT)h – h2 = 2(R + (R – R)KhK / hT)h – h2 = 2Rh – h2
(4)
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Figure 10. Rounded cone graphic analysis.
ROUNDED CONE AREA FUNCTION
SERIAL NUMBER:
DATE:
INITIALS:
APEX RAD . RP :
0.487
5/26/2008
TRANSITION RAD. RT :
2.391
BM
TRASITION DEPTH hT :
2.200
FACTOR K :
0.894
VR13211
K
2
r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h
62.3
rt:
2.049
APEX DIST. a:
1.195
MEASURED
ROUNDED SECTION
2
CONE ANGLE 2α:
CONICAL SECTION
A=πr
2
r = Tan α (a + h) A = π r2
INDENTATION DEPTH
CALCULATED RADIUS
CALCULATED AREA
INDENTATION DEPTH
CALCULATED RADIUS
hµ
rµ
A µ2
hµ
rµ
CALCULATED AREA
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
0.327
0.478
0.600
0.709
0.808
0.901
0.990
1.074
1.155
1.233
1.310
1.384
1.456
1.527
1.596
1.664
1.730
1.796
1.860
1.923
1.986
2.047
0.336628
0.716947
1.132313
1.578472
2.052567
2.552455
3.076430
3.623077
4.191191
4.779722
5.387744
6.014429
6.659027
7.320855
7.999286
8.693741
9.403681
10.128605
10.868042
11.621551
12.388713
13.169135
2.200
2.300
2.400
2.500
2.600
2.700
2.800
2.900
3.000
3.100
3.200
2.052
2.112
2.173
2.233
2.294
2.354
2.415
2.475
2.536
2.596
2.656
13.228805
14.019593
14.833337
15.670034
16.529687
17.412294
18.317855
19.246372
20.197843
21.172269
22.169649
Table 1. Rounded cone projected area calculation.
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APPENDIX A
ROUNDED CONE AREA FUNCTION EQUATION DERIVATION
Consider the rounded cone indenter shown on Fig. A1. The rounded section curve starts at the transition
point T. A circle of radius RT is drawn tangent to the cone at this point with the vertical distance to the
apex P, hT . At a smaller distance from P, h3, another circle is drawn with radius R3. Similarly several
more circles are drawn at h2, h1 and hP. The smallest circle conforms to the tip such that its radius RP is
also valid at P when h = 0.
Figure A1. Circles tangential to rounded cone.
A perfectly spherical projection radius r is given by equation (4),
r2 = 2Rh – h2
(4)
This equation is not directly applicable to a rounded cone like in Fig. A1 because R is not a constant. It is
apparent that the value of the radii Rn changes with the value of h. As the distance h gets larger the
radii of the tangent circles also get larger. So R must be a function of h,
R = f(h)
(A1)
From the rounded cone geometry the following corresponding values are found,
R = RP
when h = 0
(A2)
R = RT
when h = hT
(A3)
A possible equation for R(h) could be,
R(h) = RP + Mh
(A4)
RT = RP + MhT
(A5)
M = (RT ‐ RP) / hT
(A6)
R(h) = RP + (RT – RP)h / hT
(A7)
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And substituting in equation (4),
r2 = 2(RP + (RT – RP)h / hT)h – h2
(A8)
To test this equation, a careful measurement is made of the r values at equally spaced intervals of h on
indenter VR13211 TEM image, as illustrated on Fig. A2 . For clarity, not all values are shown. All the
measured values are inserted in Table A1.
Figure A2. r versus h measurements on indenter VR13211.
The calculated values of r and A on Table A1 are derived with equation (A8). Fig. A3 shows a plot
comparison of the measured and calculated values of A. The divergence indicates that an equation to
define R(h) for a rounded cone is not exactly linear as equation (A7). A modification was tried by adding
a coefficient and exponent to h on equation (A9). Both were tested separately but it was found that
their optimal values were similar. The same value, designated K, was chosen for exponent and
coefficient,
R(h) = RP + (RT – RP)KhK / hT
(A9)
r2 = 2(RP + (RT – RP)KhK / hT)h ‐ h2
(21)
Table A2 uses equation (21) to calculate r and A from the measured values. Fig. 4A shows the plot.
K was adjusted to the value 0.894 as shown. To find the adjusted optimal value for a particular rounded
cone only the measured value of rT is needed. Therefore only the values shown on Fig. 10 are needed to
generate the Table 1, on page 8.
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RO
OUNDED CONE AREA FUNC
CTION ‐ TEST
SERIAL NUMBER:
APEX RAD . RP :
0.487
7
5/26/2008
TRAN
NSITION RAD. RT :
1
2.391
BM
TRASSITION DEPTH hT :
0
2.200
VR1321
11
DATE:
INITIALS:
ROUN
NDED SECTION
N
2
r = 2(Rt + (Rt ‐R
Rp)h/hT)h ‐ h2
A = π r2
INDEN
NTATION DEPTH
R
MEASURED RADIUS
CALC
CULATED RADIUS
MEASURED
D AREA
ALCULATED AREA
CA
hµ
rm µ
rµ
Am µ2
µ
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
0.299
0.447
0.580
0.694
0.794
0.886
0.972
1.053
1.132
1.210
1.287
1.363
1.437
1.510
1.581
1.651
1.720
1.786
1.853
1.918
1.983
2.049
0.324
0.473
0.598
0.712
0.818
0.921
1.020
1.117
1.212
1.306
1.398
1.490
1.582
1.672
1.762
1.852
1.941
2.030
2.119
2.207
2.295
2.383
0.2808
862
0.6277
718
1.0568
832
1.5131
104
1.9805
573
2.4661
138
2.9681
126
3.4834
426
4.0257
712
4.5996
606
5.2036
637
5.8363
353
6.4872
291
7.1631
145
7.8526
602
8.5633
356
9.2940
088
10.0210
040
10.7870
001
11.5570
052
12.3536
650
13.1896
666
0.328953
0.703831
1.124633
1.591359
2.104010
2.662585
3.267085
3.917509
4.613857
5.356130
6.144327
6.978448
7.858494
8.784464
9.756359
10.774178
11.837921
12.947589
14.103181
15.304697
16.552138
17.845503
Table A1. Measured
M
and calculated vaalues of r and
d A using equaation (A8), wiithout K
Figgure A3. Plot of
o measured and calculateed values of A,
A without K
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RO
OUNDED CONE AREA FUNC
CTION ‐ TEST
SERIAL NUMBER:
APEX RAD . RP :
0.487
7
5/26/2008
TRAN
NSITION RAD. RT :
1
2.391
BM
TRASSITION DEPTH hT :
0
2.200
FACTOR K :
4
0.894
VR1321
11
DATE:
INITIALS:
ROUN
NDED SECTION
N
2
r = 2(Rt + (Rt ‐R
Rp)KhK/hT)h ‐ h2
A = π r2
INDEN
NTATION DEPTH
R
MEASURED RADIUS
CALC
CULATED RADIUS
MEASURED
D AREA
ALCULATED AREA
CA
hµ
rm µ
rµ
Am µ2
µ
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
0.299
0.447
0.580
0.694
0.794
0.886
0.972
1.053
1.132
1.210
1.287
1.363
1.437
1.510
1.581
1.651
1.720
1.786
1.853
1.918
1.983
2.049
0.327
0.478
0.600
0.709
0.808
0.901
0.990
1.074
1.155
1.233
1.310
1.384
1.456
1.527
1.596
1.664
1.730
1.796
1.860
1.923
1.986
2.047
0.2808
862
0.6277
718
1.0568
832
1.5131
104
1.9805
573
2.4661
138
2.9681
126
3.4834
426
4.0257
712
4.5996
606
5.2036
637
5.8363
353
6.4872
291
7.1631
145
7.8526
602
8.5633
356
9.2940
088
10.0210
040
10.7870
001
11.5570
052
12.3536
650
13.1896
666
0.336628
0.716947
1.132313
1.578472
2.052567
2.552455
3.076430
3.623077
4.191191
4.779722
5.387744
6.014429
6.659027
7.320855
7.999286
8.693741
9.403681
10.128605
10.868042
11.621551
12.388713
13.169135
T
Table
A2. Meaasured and caalculated valu
ues of r and A using equatiion (21), with K = 0.894
Figurre A4. Plot of measured an
nd calculated values of A, with
w K = 0.894
4
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SPHERICAL CONE TEST
To confirm the validity of equation (21), a theoretical spherical cone is drawn on Fig. A5. The
dimensions are tested on Table A3. Fig. A6 plots the comparison of measured and calculated values of
A, which are identical. The value of K is irrelevant since (RT ‐ RP) = 0. Table A4 is the complete area
function calculation for the spherical cone based on equations (21) and (22).
Figure A5. Spherical cone measurements
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RO
OUNDED CONE AREA FUNC
CTION ‐ TEST
SEERIAL NUMBER:
SPHRCON
APEX RAD . RP :
1.74
44
DATE:
5/29/200
08
TRAN
NSITION RAD. RT :
44
1.74
BM
TRASSITION DEPTH hT :
07
1.00
FACTOR K :
1.00
00
INITIALS:
ROUN
NDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2
A = π r2
INDENTTATION DEPTH
MEASURED RADIUS
CALC
CULATED RADIUSS
MEASURED AREA
ALCULATED AREA
A
CA
hµ
rm µ
rµ
µ
Am µ2
A µ2
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
1.007
0.415
0.582
0.708
0.811
0.900
0.978
1.048
1.111
1.169
1.222
1.271
1.316
1.358
1.397
1.433
1.466
1.497
1.526
1.553
1.577
1.580
0.415
0.582
0.708
0.811
0.900
0.978
1.048
1.111
1.169
1.222
1.271
1.316
1.358
1.397
1.433
1.466
1.497
1.526
1.553
1.577
1.581
0.5410
061
1.064133
1.5747
767
2.0662
291
2.5446
690
3.0048
883
3.4504
424
3.8777
734
4.293178
290
4.6912
5.0750
058
5.4407
786
5.7936
612
6.131160
6.4512
226
6.7517
773
7.0403
337
7.3157
751
7.5769
921
7.8129
918
7.8426
672
0.540040
1.064372
1.572995
2.065911
2.543119
3.004619
3.450411
3.880495
4.294871
4.693539
5.076500
5.443752
5.795296
6.131132
6.451261
6.755681
7.044393
7.317398
7.574694
7.816283
7.848851
Table A3. Measured an
nd calculated
d values of r and A for perffect spherical cone
Figgure A6. Spheerical cone plo
ot of measureed and calculated values
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ROUNDED CONE AREA FUNCTION
SERIAL NUMBER:
SPHRCON
APEX RAD . RP :
1.744
DATE:
5/29/2008
TRANSITION RAD. RT :
1.744
BM
TRASITION DEPTH hT :
1.007
FACTOR K :
1.000
INITIALS:
K
r = 2(Rp + (Rt ‐Rp)Kh /hT)h ‐ h
2
50.0
rt:
1.580
APEX DIST. a:
2.382
MEASURED
ROUNDED SECTION
2
CONE ANGLE 2α:
CONICAL SECTION
A=πr
2
r = Tan α (a + h) A = π r2
INDENTATION DEPTH
CALCULATED RADIUS
CALCULATED AREA
INDENTATION DEPTH
CALCULATED RADIUS
CALCULATED AREA
hµ
rµ
A µ2
hµ
rµ
A µ2
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000
1.007
0.415
0.582
0.708
0.811
0.900
0.978
1.048
1.111
1.169
1.222
1.271
1.316
1.358
1.397
1.433
1.466
1.497
1.526
1.553
1.577
1.581
0.540040
1.064372
1.572995
2.065911
2.543119
3.004619
3.450411
3.880495
4.294871
4.693539
5.076500
5.443752
5.795296
6.131132
6.451261
6.755681
7.044393
7.317398
7.574694
7.816283
7.848851
1.007
1.050
1.100
1.150
1.200
1.250
1.300
1.350
1.400
1.450
1.500
1.580
1.600
1.624
1.647
1.670
1.694
1.717
1.740
1.764
1.787
1.810
7.845816
8.046176
8.282329
8.521899
8.764883
9.011283
9.261099
9.514331
9.770978
10.031040
10.294518
Table A4. Spherical cone complete area function calculation.
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APPENDIX B
AREA FUNCTION EQUATION TESTS
Following is the complete set of data for three indenters analyzed with equation (21) and graphically
measured to test the equation’s validity.
ROUNDED CONE INDENTER VR13211
The data is already presented in the previous pages but is repeated here for easier access.
Figure B1. Original TEM image and basic graphics
Figure B2. r versus h measurements.
16
Micro Star Technologies Inc.
www.microstartech.com
ROUNDED CONE INDENTER VR13211
ROUNDED CONE AREA FUNCTION ‐ TEST
SERIAL NUMBER:
APEX RAD . RP :
0.487
5/26/2008
TRANSITION RAD. RT :
2.391
BM
TRASITION DEPTH hT :
2.200
FACTOR K :
0.894
VR13211
DATE:
INITIALS:
ROUNDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2
A = π r2
INDENTATION DEPTH
MEASURED RADIUS
CALCULATED RADIUS
MEASURED AREA
CALCULATED AREA
hµ
rm µ
rµ
Am µ2
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
0.299
0.447
0.580
0.694
0.794
0.886
0.972
1.053
1.132
1.210
1.287
1.363
1.437
1.510
1.581
1.651
1.720
1.786
1.853
1.918
1.983
2.049
0.327
0.478
0.600
0.709
0.808
0.901
0.990
1.074
1.155
1.233
1.310
1.384
1.456
1.527
1.596
1.664
1.730
1.796
1.860
1.923
1.986
2.047
0.280862
0.627718
1.056832
1.513104
1.980573
2.466138
2.968126
3.483426
4.025712
4.599606
5.203637
5.836353
6.487291
7.163145
7.852602
8.563356
9.294088
10.021040
10.787001
11.557052
12.353650
13.189666
0.336628
0.716947
1.132313
1.578472
2.052567
2.552455
3.076430
3.623077
4.191191
4.779722
5.387744
6.014429
6.659027
7.320855
7.999286
8.693741
9.403681
10.128605
10.868042
11.621551
12.388713
13.169135
Table B1. Measured and calculated values of r and A using equation (21), K = 0.894
17
Micro Star Technologies Inc.
www.microstartech.com
OUNDED CON
NE INDENTER
R VR13211
RO
Figure B3. Plot of meassured and calcculated valuees of A
ROUNDED CONE
C
AREA FUN
NCTION
SERIA
AL NUMBER:
DATE:
INITIALS:
APEX RAD
D . RP :
VR13211
0..487
5/26/2008
TRANSITION RAD
D. RT :
2..391
BM
TRASITION DEPTTH hT :
2..200
FACTOR K :
0..894
K
r = 2(Rp + (Rt ‐R
Rp)Kh /hT)h ‐ h
2
62.3
r t:
2.049
APEX DIST. a:
a
1.195
MEASURED
ROUN
NDED SECTION
2
CONE ANGLE 2α
α:
CO
ONICAL SECTION
N
A=πr
2
r = Tan
n α (a + h) A = π r2
INDENTATIO
ON DEPTH
CALC
CULATED RADIUS
CALCULATED AREA
INDENTATTION DEPTH
CALCULATED RADIUS
hµ
rµ
A µ2
hµ
rµ
A µ2
0.10
00
0.20
00
0.30
00
0.40
00
0.50
00
0.60
00
0.70
00
0.80
00
0.90
00
1.00
00
1.10
00
1.20
00
1.30
00
1.40
00
1.50
00
1.60
00
1.70
00
1.80
00
1.90
00
2.00
00
2.10
00
2.20
00
0.327
0.478
0.600
0.709
0.808
0.901
0.990
1.074
1.155
1.233
1.310
1.384
1.456
1.527
1.596
1.664
1.730
1.796
1.860
1.923
1.986
2.047
0.336628
0.716947
1.132313
1.578472
2.052567
2.552455
3.076430
3.623077
4.191191
4.779722
5.387744
6.014429
6.659027
7.320855
7.999286
8.693741
9.403681
10.128605
10.868042
11.621551
12.388713
13.169135
2..200
2..300
2..400
2..500
2..600
2..700
2..800
2..900
3..000
3..100
3..200
2.052
2.112
2.173
2.233
2.294
2.354
2.415
2.475
2.536
2.596
2.656
13.228805
14.019593
14.833337
15.670034
16.529687
17.412294
18.317855
19.246372
20.197843
21.172269
22.169649
CALCULATED AREA
Table
T
B2. Rou
unded cone indenter projeected area caalculation.
18
Micro Star Technologie
es Inc.
www.m
microstartech
h.com
ROUNDED CONE INDENTER VR13212
Figure B4. Original TEM image and basic graphics
Figure B5. r versus h measurements.
19
Micro Star Technologies Inc.
www.microstartech.com
ROUNDED CONE INDENTER VR13212
ROUNDED CONE AREA FUNCTION ‐ TEST
SERIAL NUMBER:
APEX RAD . RP :
0.325
5/26/2008
TRANSITION RAD. RT :
2.201
BM
TRASITION DEPTH hT :
2.600
FACTOR K :
0.945
VR13212
DATE:
INITIALS:
ROUNDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2
A = π r2
INDENTATION DEPTH
MEASURED RADIUS
CALCULATED RADIUS
MEASURED AREA
CALCULATED AREA
hµ
rm µ
rµ
Am µ2
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
2.300
2.400
2.500
2.600
0.218
0.354
0.463
0.558
0.641
0.717
0.788
0.856
0.923
0.989
1.055
1.119
1.181
1.241
1.300
1.357
1.414
1.470
1.526
1.582
1.637
1.693
1.749
1.805
1.861
1.917
0.265
0.387
0.486
0.574
0.655
0.731
0.804
0.874
0.941
1.007
1.071
1.133
1.194
1.255
1.314
1.372
1.429
1.486
1.542
1.597
1.652
1.706
1.760
1.813
1.865
1.917
0.149301
0.393692
0.673460
0.978179
1.290821
1.615058
1.950753
2.301958
2.676414
3.072858
3.496671
3.933780
4.381771
4.838307
5.309292
5.785083
6.281288
6.788668
7.315751
7.862539
8.418743
9.004587
9.610135
10.235387
10.880344
11.545004
0.221414
0.469973
0.741843
1.035064
1.348293
1.680511
2.030897
2.398764
2.783523
3.184657
3.601707
4.034263
4.481948
4.944421
5.421364
5.912486
6.417513
6.936189
7.468274
8.013542
8.571779
9.142781
9.726355
10.322317
10.930490
11.550708
Table B3. Measured and calculated values of r and A using equation (21), K = 0.945
20
Micro Star Technologies Inc.
www.microstartech.com
OUNDED CON
NE INDENTER
R VR13212
RO
Figure B6. Plot of meassured and calcculated valuees of A
ROUNDED CONE
C
AREA FUN
NCTION
SERIAL NUMBER:
DATE:
INITIALS:
APEX RAD
D . RP :
0.325
5/26/2008
TRANSITION RAD
D. RT :
2.201
BM
TRASITION DEPTTH hT :
2.600
FACTOR K :
0.945
VR13212
K
2
r = 2(Rp + (Rtt ‐Rp)Kh /hT)h ‐ h
54.6
rt:
1.917
APEX DIST. a:
1.131
MEASURED
RO
OUNDED SECTION
N
2
C
CONE
ANGLE 2α:
CONICAL SECTION
A=πr
2
r = Tan α (a + h) A = π r2
INDENTTATION DEPTH
CA
ALCULATED RADIUS
REA
CALCULATED AR
INDENTATION DEPTH
CALCU
ULATED RADIUS
CALCULATED AREA
hµ
rµ
A µ2
hµ
rµ
A µ2
0.100
0.200
0.300
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
1.200
1.300
1.400
1.500
1.600
1.700
1.800
1.900
2.000
2.100
2.200
2.300
2.400
2.500
2.600
0.265
0.387
0.486
0.574
0.655
0.731
0.804
0.874
0.941
1.007
1.071
1.133
1.194
1.255
1.314
1.372
1.429
1.486
1.542
1.597
1.652
1.706
1.760
1.813
1.865
1.917
0.221414
0.469973
0.741843
1.035064
1.348293
1.680511
2.030897
2.398764
2.783523
3.184657
3.601707
4.034263
4.481948
4.944421
5.421364
5.912486
6.417513
6.936189
7.468274
8.013542
8.571779
9.142781
9.726355
10.322317
10.930490
11.550708
2.600
2.700
2.800
2.900
3.000
3.100
3.200
3.300
3.400
3.500
3.600
1.926
1.977
2.029
2.081
2.132
2.184
2.235
2.287
2.339
2.390
2.442
11.650186
12.283063
12.932678
13.599031
14.282122
14.981952
15.698521
16.431827
17.181872
17.948656
18.732177
Table
T
B4. Rou
unded cone indenter projeected area caalculation.
21
Micro Star Technologie
es Inc.
www.m
microstartech
h.com
ROUNDED CONE INDENTER VR13240
Figure B7. Original TEM image and basic graphics
Figure B8. r versus h measurements.
22
Micro Star Technologies Inc.
www.microstartech.com
ROUNDED CONE INDENTER VR13240
ROUNDED CONE AREA FUNCTION ‐ TEST
APEX RAD . RP :
0.875
5/26/2008
TRANSITION RAD. RT :
1.596
BM
TRASITION DEPTH hT :
0.850
FACTOR K :
0.765
SERIAL NUMBER:
DATE:
VR13240
INITIALS:
ROUNDED SECTION
2
r = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2
A = π r2
INDENTATION DEPTH
MEASURED RADIUS
CALCULATED RADIUS
MEASURED AREA
CALCULATED AREA
hµ
rm µ
rµ
Am µ2
A µ2
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.283
0.412
0.514
0.604
0.689
0.766
0.833
0.892
0.947
1.000
1.051
1.101
1.149
1.194
1.237
1.278
1.318
0.303
0.433
0.534
0.621
0.698
0.768
0.833
0.893
0.950
1.003
1.054
1.103
1.150
1.194
1.237
1.279
1.319
0.251607
0.533267
0.829996
1.146103
1.491380
1.843348
2.179917
2.499652
2.817409
3.141593
3.470206
3.808242
4.147534
4.478768
4.807168
5.131113
5.457336
0.287644
0.588405
0.897253
1.211948
1.531054
1.853534
2.178582
2.505544
2.833874
3.163104
3.492828
3.822686
4.152356
4.481550
4.810004
5.137476
5.463746
Table B5. Measured and calculated values of r and A using equation (21), K = 0.765
23
Micro Star Technologies Inc.
www.microstartech.com
OUNDED CON
NE INDENTER
R VR13240
RO
Figure B9. Plot of meassured and calcculated valuees of A
ROUNDED CONE
C
AREA FUN
NCTION
SERIA
AL NUMBER:
DATE:
INITIALS:
APEX RAD
D . RP :
0..875
5/28/2008
TRANSITION RAD
D. RT :
1..596
BM
TRASITION DEPTTH hT :
0..850
FACTOR K :
0..765
VR13240
K
r = 2(Rp + (Rt ‐R
Rp)Kh /hT)h ‐ h
2
68.6
rt:
1.319
APEX DIST. a:
a
1.086
MEASURED
ROUN
NDED SECTION
2
CONE ANGLE 2α
α:
CO
ONICAL SECTION
N
A=πr
2
n α (a + h) A = π r2
r = Tan
INDENTATIO
ON DEPTH
CALC
CULATED RADIUS
CALCULATED AREA
INDENTATTION DEPTH
CALCULATED RADIUS
hµ
rµ
A µ2
hµ
rµ
A µ2
0.05
50
0.10
00
0.15
50
0.20
00
0.25
50
0.30
00
0.35
50
0.40
00
0.45
50
0.50
00
0.55
50
0.60
00
0.65
50
0.70
00
0.75
50
0.80
00
0.85
50
0.303
0.433
0.534
0.621
0.698
0.768
0.833
0.893
0.950
1.003
1.054
1.103
1.150
1.194
1.237
1.279
1.319
0.287644
0.588405
0.897253
1.211948
1.531054
1.853534
2.178582
2.505544
2.833874
3.163104
3.492828
3.822686
4.152356
4.481550
4.810004
5.137476
5.463746
0..850
0..900
0..950
1..000
1..050
1..100
1..150
1..200
1..250
1..300
1..350
1.321
1.355
1.389
1.423
1.457
1.491
1.525
1.559
1.594
1.628
1.662
1
5.479301
5.765977
7
6.059963
3
6.361258
8
6.669863
3
6.985777
7
7.309001
1
7.639534
4
7.977377
7
8.322529
9
8.674990
0
CALCULATED AREA
Table
T
B6. Rou
unded cone indenter projeected area caalculation.
24
Micro Star Technologie
es Inc.
www.m
microstartech
h.com