spherical and rounded cone nano indentersmicrostartech.com/index/conicalindenters2.pdf · 1 micro...

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.


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)


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


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)


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




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.


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)


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


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)


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


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


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


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.



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







FACTOR K : 0.945

INDENTATION DEPTH

h µ

MEASURED 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



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







FACTOR K : 0.765

INDENTATION DEPTH

h µ

MEASURED 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



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

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