1 Micro Star Technologies Inc. www.microstartech.com
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.
2 Micro Star Technologies Inc. www.microstartech.com
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 Sphere radius h Indentation depth r Radius of projected circle at indentation depth α Cone half angle T Transition between cone and sphere C Sphere center P Indenter apex
O Cone theoretical apex a 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) (1)
A = π r2 (2)
3 Micro Star Technologies Inc. www.microstartech.com
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
4 Micro Star Technologies Inc. www.microstartech.com
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)
5 Micro Star Technologies Inc. www.microstartech.com
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
6 Micro Star Technologies Inc. www.microstartech.com
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 Apex circle radius RT Transition point circle radius hT Indentation depth at transition rT Projected radius at transition depth K Adjustable h coefficient and exponent.
7 Micro Star Technologies Inc. www.microstartech.com
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)
8 Micro Star Technologies Inc. www.microstartech.com
Figure 10. Rounded cone graphic analysis.
Table 1. Rounded cone projected area calculation.
SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487 CONE ANGLE 2α: 62.3
DATE: 5/26/2008 TRANSITION RAD. RT : 2.391 MEASURED rt: 2.049
INITIALS: BM TRASITION DEPTH hT : 2.200 APEX DIST. a: 1.195
FACTOR K : 0.894
INDENTATION DEPTH
h µ
CALCULATED RADIUS
r µCALCULATED AREA
A µ2
INDENTATION DEPTH
h µ
CALCULATED RADIUS
r µCALCULATED AREA
A µ2
0.100 0.327 0.336628 2.200 2.052 13.2288050.200 0.478 0.716947 2.300 2.112 14.0195930.300 0.600 1.132313 2.400 2.173 14.8333370.400 0.709 1.578472 2.500 2.233 15.6700340.500 0.808 2.052567 2.600 2.294 16.5296870.600 0.901 2.552455 2.700 2.354 17.4122940.700 0.990 3.076430 2.800 2.415 18.3178550.800 1.074 3.623077 2.900 2.475 19.2463720.900 1.155 4.191191 3.000 2.536 20.1978431.000 1.233 4.779722 3.100 2.596 21.1722691.100 1.310 5.387744 3.200 2.656 22.1696491.200 1.384 6.0144291.300 1.456 6.6590271.400 1.527 7.3208551.500 1.596 7.9992861.600 1.664 8.6937411.700 1.730 9.4036811.800 1.796 10.1286051.900 1.860 10.8680422.000 1.923 11.6215512.100 1.986 12.3887132.200 2.047 13.169135
ROUNDED SECTION CONICAL SECTION
r2 = 2(Rp + (Rt ‐Rp)KhK/hT)h ‐ h2 A = π r2
ROUNDED CONE AREA FUNCTION
r = Tan α (a + h) A = π r2
9 Micro Star Technologies Inc. www.microstartech.com
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)
10 Micro Star Technologies Inc. www.microstartech.com
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.
Micro Sta
INDEN
r Technologie
Table A1. M
Fig
SERIAL NUMBER
DATE
INITIALS
NTATION DEPTH
h µ
0.1000.2000.3000.4000.5000.6000.7000.8000.9001.0001.1001.2001.3001.4001.5001.6001.7001.8001.9002.0002.1002.200
es Inc.
Measured and
gure A3. Plot o
: VR1321
: 5/26/20
: BM
MEASURED R
rm µ
0.2990.4470.5800.6940.7940.8860.9721.0531.1321.2101.2871.3631.4371.5101.5811.6511.7201.7861.8531.9181.9832.049
r2
RO
www.m
calculated va
of measured
11
08 TRAN
TRAS
RADIUS CALC
ROUN2 = 2(Rt + (Rt ‐R
OUNDED CON
microstartech
alues of r and
and calculate
APEX RAD . RP
NSITION RAD. RT
SITION DEPTH hT
CULATED RADIUS
r µ
0.3240.4730.5980.7120.8180.9211.0201.1171.2121.3061.3981.4901.5821.6721.7621.8521.9412.0302.1192.2072.2952.383
NDED SECTION
Rp)h/hT)h ‐ h2
E AREA FUNC
h.com
d A using equa
ed values of A
P : 0.487
T : 2.391
T : 2.200
MEASURED
Am µ
0.28080.62771.05681.51311.98052.46612.96813.48344.02574.59965.20365.83636.48727.16317.85268.56339.2940
10.021010.787011.557012.353613.1896
N2 A = π r2
CTION ‐ TEST
ation (A8), wi
A, without K
7
1
0
D AREA
µ2
CA
862718832104573138126426712606637353291145602356088040001052650666
ithout K
ALCULATED AREA
A µ2
0.3289530.7038311.1246331.5913592.1040102.6625853.2670853.9175094.6138575.3561306.1443276.9784487.8584948.7844649.756359
10.77417811.83792112.94758914.10318115.30469716.55213817.845503
11
Micro Sta
T
INDEN
r Technologie
Table A2. Mea
Figur
SERIAL NUMBER
DATE
INITIALS
NTATION DEPTH
h µ
0.1000.2000.3000.4000.5000.6000.7000.8000.9001.0001.1001.2001.3001.4001.5001.6001.7001.8001.9002.0002.1002.200
es Inc.
asured and ca
re A4. Plot of
: VR1321
: 5/26/20
: BM
MEASURED R
rm µ
0.2990.4470.5800.6940.7940.8860.9721.0531.1321.2101.2871.3631.4371.5101.5811.6511.7201.7861.8531.9181.9832.049
r2 =
RO
www.m
alculated valu
measured an
11
08 TRAN
TRAS
RADIUS CALC
ROUN
= 2(Rt + (Rt ‐R
OUNDED CON
microstartech
ues of r and A
nd calculated
APEX RAD . RP
NSITION RAD. RT
SITION DEPTH hT
FACTOR K
CULATED RADIUS
r µ
0.3270.4780.6000.7090.8080.9010.9901.0741.1551.2331.3101.3841.4561.5271.5961.6641.7301.7961.8601.9231.9862.047
NDED SECTION
Rp)KhK/hT)h ‐ h
E AREA FUNC
h.com
A using equati
values of A, w
P : 0.487
T : 2.391
T : 2.200
: 0.894
MEASURED
Am µ
0.28080.62771.05681.51311.98052.46612.96813.48344.02574.59965.20365.83636.48727.16317.85268.56339.2940
10.021010.787011.557012.353613.1896
N
h2 A = π r2
CTION ‐ TEST
ion (21), with
with K = 0.894
7
1
0
4
D AREA
µ2
CA
862718832104573138126426712606637353291145602356088040001052650666
K = 0.894
4
ALCULATED AREA
A µ2
0.3366280.7169471.1323131.5784722.0525672.5524553.0764303.6230774.1911914.7797225.3877446.0144296.6590277.3208557.9992868.6937419.403681
10.12860510.86804211.62155112.38871313.169135
12
13 Micro Star Technologies Inc. www.microstartech.com
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
Micro Sta
SE
INDENT
r Technologie
Table A3.
Fig
ERIAL NUMBER:
DATE:
INITIALS:
TATION DEPTH
h µ
0.0500.1000.1500.2000.2500.3000.3500.4000.4500.5000.5500.6000.6500.7000.7500.8000.8500.9000.9501.0001.007
es Inc.
Measured an
gure A6. Sphe
SPHRCO
5/29/200
BM
MEASURED R
rm µ
0.4150.5820.7080.8110.9000.9781.0481.1111.1691.2221.2711.3161.3581.3971.4331.4661.4971.5261.5531.5771.580
r2 =
RO
www.m
nd calculated
erical cone plo
N
08 TRAN
TRAS
ADIUS CALC
ROUN
= 2(Rt + (Rt ‐R
OUNDED CON
microstartech
d values of r a
ot of measure
APEX RAD . RP
NSITION RAD. R
SITION DEPTH hT
FACTOR K
CULATED RADIUS
r µ
0.4150.5820.7080.8110.9000.9781.0481.1111.1691.2221.2711.3161.3581.3971.4331.4661.4971.5261.5531.5771.581
NDED SECTIO
p)KhK/hT)h ‐ h
E AREA FUNC
h.com
nd A for perf
ed and calcul
P : 1.74
T : 1.74
T : 1.00
K : 1.00
S MEASURE
Am µ
0.54101.0641.57472.06622.54463.00483.45043.87774.2934.69125.07505.44075.79366.1316.45126.75177.04037.31577.57697.81297.8426
N
h2 A = π r2
CTION ‐ TEST
fect spherical
ated values
44
44
07
00
D AREA
µ2
CA
061133767291690883424734178290058786612160226773337751921918672
cone
ALCULATED AREA
A µ2
0.5400401.0643721.5729952.0659112.5431193.0046193.4504113.8804954.2948714.6935395.0765005.4437525.7952966.1311326.4512616.7556817.0443937.3173987.5746947.8162837.848851
14
A
15 Micro Star Technologies Inc. www.microstartech.com
Table A4. Spherical cone complete area function calculation.
SERIAL NUMBER: SPHRCON APEX RAD . RP : 1.744 CONE ANGLE 2α: 50.0
DATE: 5/29/2008 TRANSITION RAD. RT : 1.744 MEASURED rt: 1.580
INITIALS: BM TRASITION DEPTH hT : 1.007 APEX DIST. a: 2.382
FACTOR K : 1.000
INDENTATION DEPTH
h µ
CALCULATED RADIUS
r µCALCULATED AREA
A µ2
INDENTATION DEPTH
h µ
CALCULATED RADIUS
r µCALCULATED AREA
A µ2
0.050 0.415 0.540040 1.007 1.580 7.8458160.100 0.582 1.064372 1.050 1.600 8.0461760.150 0.708 1.572995 1.100 1.624 8.2823290.200 0.811 2.065911 1.150 1.647 8.5218990.250 0.900 2.543119 1.200 1.670 8.7648830.300 0.978 3.004619 1.250 1.694 9.0112830.350 1.048 3.450411 1.300 1.717 9.2610990.400 1.111 3.880495 1.350 1.740 9.5143310.450 1.169 4.294871 1.400 1.764 9.7709780.500 1.222 4.693539 1.450 1.787 10.0310400.550 1.271 5.076500 1.500 1.810 10.2945180.600 1.316 5.4437520.650 1.358 5.7952960.700 1.397 6.1311320.750 1.433 6.4512610.800 1.466 6.7556810.850 1.497 7.0443930.900 1.526 7.3173980.950 1.553 7.5746941.000 1.577 7.8162831.007 1.581 7.848851
ROUNDED SECTION CONICAL SECTION
r2 = 2(Rp + (Rt ‐Rp)KhK/hT)h ‐ h2 A = π r2
ROUNDED CONE AREA FUNCTION
r = Tan α (a + h) A = π r2
16 Micro Star Technologies Inc. www.microstartech.com
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.
17 Micro Star Technologies Inc. www.microstartech.com
ROUNDED CONE INDENTER VR13211
Table B1. Measured and calculated values of r and A using equation (21), K = 0.894
SERIAL NUMBER: VR13211 APEX RAD . RP : 0.487
DATE: 5/26/2008 TRANSITION RAD. RT : 2.391
INITIALS: BM TRASITION DEPTH hT : 2.200
FACTOR K : 0.894
INDENTATION DEPTH
h µ
MEASURED RADIUS
rm µCALCULATED RADIUS
r µMEASURED AREA
Am µ2
CALCULATED AREA
A µ2
0.100 0.299 0.327 0.280862 0.3366280.200 0.447 0.478 0.627718 0.7169470.300 0.580 0.600 1.056832 1.1323130.400 0.694 0.709 1.513104 1.5784720.500 0.794 0.808 1.980573 2.0525670.600 0.886 0.901 2.466138 2.5524550.700 0.972 0.990 2.968126 3.0764300.800 1.053 1.074 3.483426 3.6230770.900 1.132 1.155 4.025712 4.1911911.000 1.210 1.233 4.599606 4.7797221.100 1.287 1.310 5.203637 5.3877441.200 1.363 1.384 5.836353 6.0144291.300 1.437 1.456 6.487291 6.6590271.400 1.510 1.527 7.163145 7.3208551.500 1.581 1.596 7.852602 7.9992861.600 1.651 1.664 8.563356 8.6937411.700 1.720 1.730 9.294088 9.4036811.800 1.786 1.796 10.021040 10.1286051.900 1.853 1.860 10.787001 10.8680422.000 1.918 1.923 11.557052 11.6215512.100 1.983 1.986 12.353650 12.3887132.200 2.049 2.047 13.189666 13.169135
ROUNDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2 A = π r2
ROUNDED CONE AREA FUNCTION ‐ TEST
Micro Sta
SERIA
INDENTATIO
h
0.100.200.300.400.500.600.700.800.901.001.101.201.301.401.501.601.701.801.902.002.102.20
r2
r Technologie
T
AL NUMBER:
DATE:
INITIALS:
ON DEPTH
µ
CALC
00000000000000000000000000000000000000000000
ROUN2 = 2(Rp + (Rt ‐R
es Inc.
RO
Figure B3.
Table B2. Rou
VR13211
5/26/2008
BM
CULATED RADIUS
r µ
0.3270.4780.6000.7090.8080.9010.9901.0741.1551.2331.3101.3841.4561.5271.5961.6641.7301.7961.8601.9231.9862.047
NDED SECTION
Rp)KhK/hT)h ‐ h2
www.m
OUNDED CON
Plot of meas
unded cone i
APEX RAD
TRANSITION RAD
TRASITION DEPT
FACTO
CALCULATED A
A µ2
0.3366280.7169471.1323131.5784722.0525672.5524553.0764303.6230774.1911914.7797225.3877446.0144296.6590277.3208557.9992868.6937419.403681
10.12860510.86804211.62155112.38871313.169135
2 A = π r2
ROUNDED C
microstartech
NE INDENTER
sured and calc
ndenter proje
D . RP : 0.
D. RT : 2.
TH hT : 2.
R K : 0.
REA INDENTAT
h
2.2.2.2.2.2.2.2.3.3.3.
CONE AREA FUN
h.com
R VR13211
culated value
ected area ca
.487
.391
.200
.894
TION DEPTH
h µ
CA
.200
.300
.400
.500
.600
.700
.800
.900
.000
.100
.200
CO
NCTION
r = Tan
es of A
alculation.
CONE ANGLE 2α
MEASURED rAPEX DIST. a
LCULATED RADIUS
r µ
2.0522.1122.1732.2332.2942.3542.4152.4752.5362.5962.656
ONICAL SECTION
n α (a + h) A =
α: 62.3
rt: 2.049
a: 1.195
CALCULATED
A µ2
13.2288014.0195914.8333315.6700316.5296817.4122918.3178519.2463720.1978421.1722622.16964
N
π r2
18
AREA
53747452399
19 Micro Star Technologies Inc. www.microstartech.com
ROUNDED CONE INDENTER VR13212
Figure B4. Original TEM image and basic graphics
Figure B5. r versus h measurements.
20 Micro Star Technologies Inc. www.microstartech.com
ROUNDED CONE INDENTER VR13212
Table B3. Measured and calculated values of r and A using equation (21), K = 0.945
SERIAL NUMBER: VR13212 APEX RAD . RP : 0.325
DATE: 5/26/2008 TRANSITION RAD. RT : 2.201
INITIALS: BM TRASITION DEPTH hT : 2.600
FACTOR K : 0.945
INDENTATION DEPTH
h µ
MEASURED RADIUS
rm µCALCULATED RADIUS
r µMEASURED AREA
Am µ2
CALCULATED AREA
A µ2
0.100 0.218 0.265 0.149301 0.2214140.200 0.354 0.387 0.393692 0.4699730.300 0.463 0.486 0.673460 0.7418430.400 0.558 0.574 0.978179 1.0350640.500 0.641 0.655 1.290821 1.3482930.600 0.717 0.731 1.615058 1.6805110.700 0.788 0.804 1.950753 2.0308970.800 0.856 0.874 2.301958 2.3987640.900 0.923 0.941 2.676414 2.7835231.000 0.989 1.007 3.072858 3.1846571.100 1.055 1.071 3.496671 3.6017071.200 1.119 1.133 3.933780 4.0342631.300 1.181 1.194 4.381771 4.4819481.400 1.241 1.255 4.838307 4.9444211.500 1.300 1.314 5.309292 5.4213641.600 1.357 1.372 5.785083 5.9124861.700 1.414 1.429 6.281288 6.4175131.800 1.470 1.486 6.788668 6.9361891.900 1.526 1.542 7.315751 7.4682742.000 1.582 1.597 7.862539 8.0135422.100 1.637 1.652 8.418743 8.5717792.200 1.693 1.706 9.004587 9.1427812.300 1.749 1.760 9.610135 9.7263552.400 1.805 1.813 10.235387 10.3223172.500 1.861 1.865 10.880344 10.9304902.600 1.917 1.917 11.545004 11.550708
ROUNDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2 A = π r2
ROUNDED CONE AREA FUNCTION ‐ TEST
Micro Sta
S
INDENT
r Technologie
T
ERIAL NUMBER:
DATE:
INITIALS:
TATION DEPTH
h µ
CA
0.1000.2000.3000.4000.5000.6000.7000.8000.9001.0001.1001.2001.3001.4001.5001.6001.7001.8001.9002.0002.1002.2002.3002.4002.5002.600
RO
r2 = 2(Rp + (Rt
es Inc.
RO
Figure B6.
Table B4. Rou
VR13212
5/26/2008
BM
ALCULATED RADIUS
r µ
0.2650.3870.4860.5740.6550.7310.8040.8740.9411.0071.0711.1331.1941.2551.3141.3721.4291.4861.5421.5971.6521.7061.7601.8131.8651.917
OUNDED SECTION
t ‐Rp)KhK/hT)h ‐ h
www.m
OUNDED CON
Plot of meas
unded cone i
APEX RAD
TRANSITION RAD
TRASITION DEPT
FACTO
CALCULATED AR
A µ2
0.2214140.4699730.7418431.0350641.3482931.6805112.0308972.3987642.7835233.1846573.6017074.0342634.4819484.9444215.4213645.9124866.4175136.9361897.4682748.0135428.5717799.1427819.726355
10.32231710.93049011.550708
N
h2 A = π r2
ROUNDED C
microstartech
NE INDENTER
sured and calc
ndenter proje
D . RP : 0.3
D. RT : 2.2
TH hT : 2.6
R K : 0.9
REA INDENTATI
h
2.62.72.82.93.03.13.23.33.43.53.6
CONE AREA FUN
h.com
R VR13212
culated value
ected area ca
25 C
01
00
45
ON DEPTH
µ
CALCU
0000000000000000000000
CONI
NCTION
r = Tan α
es of A
alculation.
CONE ANGLE 2α:
MEASURED rt:
APEX DIST. a:
ULATED RADIUS
r µ
1.9261.9772.0292.0812.1322.1842.2352.2872.3392.3902.442
CAL SECTION
α (a + h) A = π r2
54.6
1.917
1.131
CALCULATED AREA
A µ2
11.65018612.28306312.93267813.59903114.28212214.98195215.69852116.43182717.18187217.94865618.732177
2
21
22 Micro Star Technologies Inc. www.microstartech.com
ROUNDED CONE INDENTER VR13240
Figure B7. Original TEM image and basic graphics
Figure B8. r versus h measurements.
23 Micro Star Technologies Inc. www.microstartech.com
ROUNDED CONE INDENTER VR13240
Table B5. Measured and calculated values of r and A using equation (21), K = 0.765
SERIAL NUMBER: VR13240 APEX RAD . RP : 0.875
DATE: 5/26/2008 TRANSITION RAD. RT : 1.596
INITIALS: BM TRASITION DEPTH hT : 0.850
FACTOR K : 0.765
INDENTATION DEPTH
h µ
MEASURED RADIUS
rm µCALCULATED RADIUS
r µMEASURED AREA
Am µ2
CALCULATED AREA
A µ2
0.050 0.283 0.303 0.251607 0.2876440.100 0.412 0.433 0.533267 0.5884050.150 0.514 0.534 0.829996 0.8972530.200 0.604 0.621 1.146103 1.2119480.250 0.689 0.698 1.491380 1.5310540.300 0.766 0.768 1.843348 1.8535340.350 0.833 0.833 2.179917 2.1785820.400 0.892 0.893 2.499652 2.5055440.450 0.947 0.950 2.817409 2.8338740.500 1.000 1.003 3.141593 3.1631040.550 1.051 1.054 3.470206 3.4928280.600 1.101 1.103 3.808242 3.8226860.650 1.149 1.150 4.147534 4.1523560.700 1.194 1.194 4.478768 4.4815500.750 1.237 1.237 4.807168 4.8100040.800 1.278 1.279 5.131113 5.1374760.850 1.318 1.319 5.457336 5.463746
ROUNDED SECTION
r2 = 2(Rt + (Rt ‐Rp)KhK/hT)h ‐ h2 A = π r2
ROUNDED CONE AREA FUNCTION ‐ TEST
Micro Sta
SERIA
INDENTATIO
h
0.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.85
r2
r Technologie
T
AL NUMBER:
DATE:
INITIALS:
ON DEPTH
µ
CALC
5000500050005000500050005000500050
ROUN2 = 2(Rp + (Rt ‐R
es Inc.
RO
Figure B9.
Table B6. Rou
VR13240
5/28/2008
BM
CULATED RADIUS
r µ
0.3030.4330.5340.6210.6980.7680.8330.8930.9501.0031.0541.1031.1501.1941.2371.2791.319
NDED SECTION
Rp)KhK/hT)h ‐ h2
www.m
OUNDED CON
Plot of meas
unded cone i
APEX RAD
TRANSITION RAD
TRASITION DEPT
FACTO
CALCULATED A
A µ2
0.2876440.5884050.8972531.2119481.5310541.8535342.1785822.5055442.8338743.1631043.4928283.8226864.1523564.4815504.8100045.1374765.463746
2 A = π r2
ROUNDED C
microstartech
NE INDENTER
sured and calc
ndenter proje
D . RP : 0.
D. RT : 1.
TH hT : 0.
R K : 0.
REA INDENTAT
h
0.0.0.1.1.1.1.1.1.1.1.
CONE AREA FUN
h.com
R VR13240
culated value
ected area ca
.875
.596
.850
.765
TION DEPTH
h µ
CA
.850
.900
.950
.000
.050
.100
.150
.200
.250
.300
.350
CO
NCTION
r = Tan
es of A
alculation.
CONE ANGLE 2α
MEASURED rAPEX DIST. a
LCULATED RADIUS
r µ
1.3211.3551.3891.4231.4571.4911.5251.5591.5941.6281.662
ONICAL SECTION
n α (a + h) A =
α: 68.6
rt: 1.319
a: 1.086
CALCULATED
A µ2
5.4793015.7659776.0599636.3612586.6698636.9857777.3090017.6395347.9773778.3225298.674990
N
π r2
24
AREA
17383714790