on the measurement of yield strength
DESCRIPTION
Nanoindentation, spherical tip, yield strength estimationTRANSCRIPT
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ON THE MEASUREMENT OF YIELD STRENGTH BY SPHERICAL INDENTATION
E. G. Herbert 1, W. C. Oliver 1, and G. M. Pharr 2
1 University of Tennessee, Dept. of Materials Science and Engineering; & MTS Nano Instruments Innovation Center
2 University of Tennessee, Dept. of Materials Science and Engineering; & Oak Ridge National Laboratory, Metals and Ceramics Division
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CONTEMPORARY INVESTIGATIONS
• 1995 Field and Swain
1996 Y d Bl h d• 1996 Yu and Blanchard
• 1998 Taljet, Zacharia, and Kosel
• 2001 Schwarzer et al.How well do they work?
• Test material: Al 6061‐T6• 2002 Durst, Goken, and Pharr
• 2003 Ma, Ong, Lu, and He
• 2004 Cao and Lu
• Test material: Al 6061 T6
• Diamond sphere, measured radius of 385 nm
( ll h i ll• 2004 Cao and Lu
• 2004 Mulford, Asaro, and Sebring
• 2004 Kogut and Komvopoulos
(smallest sphere commercially available)
• 2004 Lee and Lee
• 2004 Kwon et al.
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UNIAXIAL TENSILE MEASUREMENTS
400
l
300
350
MPa
)
Al 6061‐T6
200
250
tres
s (M
E = 72.59 GPa ± 2.54%(confirmed ultrasonically)
273 MP ± 0 70%
100
150
True
St = 273 MPa ± 0.70%yσ
Literature Values:E = 69 GPa
0
50
0 0 02 0 04 0 06 0 08 0 1 0 12
T E 69 GPa= 275 MPayσ
0 0.02 0.04 0.06 0.08 0.1 0.12True Strain (-)
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UNIAXIAL TENSILE MEASUREMENTS
400
l
300
350
MPa
)
Al 6061‐T6
200
250
tres
s (M
nkεσ =Power law fit:
100
150
True
St 093.00.432 εσ =
0
50
0 0 02 0 04 0 06 0 08 0 1 0 12
T
0 0.02 0.04 0.06 0.08 0.1 0.12True Strain (-)
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CONFIRMATION OF MEASURED TIP RADIUS
3 280
0 20 40 60 80 100M
Contact Depth (nm)
2 4
2.8
3.2
60
70
Modulus from unload,(mN
)M
odulus
1.6
2
2.4
40
50R = 385 nm
Sam
ple
s of ElaFused silica
0.8
1.2
6
20
30
oad
on S
asticity (
0
0.4
0
10
0 20 40 60 80 100 120 140 160
Experimental DataHertz Theory, R = 385 nm
Lo(G
Pa)
0 20 40 60 80 100 120 140 160Displacement Into Surface (nm)
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INDENTATION DATA
0.5N
)
0 3
0.4
ple
(mN
Controlled loading,
Al 6061‐T6
0 2
0.3
n Sa
mp g,
P/P = 0.05 s‐1.
0.1
0.2
oad
On
00 20 40 60 80 100
Lo
0 20 40 60 80 100Displacement Into Surface (nm)
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EFFECTS OF SURFACE ROUGHNESS
Mechanically polished Al 6061‐T6Al 6061 T6
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EFFECTS OF SURFACE ROUGHNESS
Mechanically polished Al 6061‐T6Al 6061 T6
Prevents us from accuratelydetermining small stressesgand strains.
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Ma et al., J. Appl. Phys. 94, 1 (2003)
X
m hhPP ⎟⎟⎠
⎞⎜⎜⎝
⎛= ⎟⎟
⎠
⎞⎜⎜⎝
⎛Φ=
Rh
EEn
EERP miyH
Hm ,,,2
σ⎟⎟⎠
⎞⎜⎜⎝
⎛=
Rh
EEn
EX miyH
H ,,,σ
ψmh ⎠⎝ ⎠⎝ REEER ⎠⎝
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Ma et al., J. Appl. Phys. 94, 1 (2003)
1.00 < X < 1.36
0 04
hm = 0.05R = 19.25 nm
hm = 0.025R = 9.63 nm
hm = 0.01R = 3.85 nm
0.001 < Pm / (ER2) < 0.01
Pm / (ER2):0 03
0.035
0.04
mN
) P = 0.03666(h/hm)1.8417
P = 0.0097747(h/h )1.7786
0.01R→ 0.0002110.025R→ 0.0009080.05R→ 0.003407
0 02
0.025
0.03
ampl
e (m (
m)
P = 2.2654E-3(h/hm)1.2192
0 01
0.015
0.02
d O
n Sa
This isn’t working!Presumably due to
0
0.005
0.01
Loa Presumably due to
roughness and contaminants on thesurface.0
0 0.2 0.4 0.6 0.8 1h/h
m (-)
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Cao and Lu, Acta Materialia 52, (2004)n
fy
yE
⎟⎟⎠
⎞⎜⎜⎝
⎛+= εσ
σσ 1 ⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛= 43
22
31
2 lnlnln CECECEChPr
r
r
r
r
rgrg σσσ
σ
2
106.1435.000939.0 ⎟⎟⎞
⎜⎜⎛
−+=Rh
Rh gg
rε
n
⎟⎠
⎜⎝ RRr
n
ry
yrE
⎥⎥⎦
⎤
⎢⎢⎣
⎡+= 1,1, 1 εσ
σσ
n
ry
yrE
⎥⎥⎦
⎤
⎢⎢⎣
⎡+= 2,2, 1 εσ
σσy ⎦⎣
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Cao and Lu, Acta Materialia 52, (2004)
⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛= 43
22
31
2 lnlnln CECECEChPr
r
r
r
r
rgrg σσσ
σn
fy
yE
⎟⎟⎠
⎞⎜⎜⎝
⎛+= εσ
σσ 1
hg, 1 = ~ 0.01R = 4.2 nm
P 1 = 2 18 µN
2
106.1435.000939.0 ⎟⎟⎞
⎜⎜⎛
−+=Rh
Rh gg
rεPg, 1 2.18 µN
hg, 2 = ~ 0.06R = 23.3 nm
P 52 98 N n
⎟⎠
⎜⎝ RRr
Pg, 2 = 52.98 µN
= 4.24E+7 Pa1,rσSolution does notconverge for 0 ≤ n ≤ 1
n
ry
yrE
⎥⎥⎦
⎤
⎢⎢⎣
⎡+= 1,1, 1 εσ
σσ
= 2.98E+8 Pa
= 0.0141
2,rσ
1,rε
0 ≤ n ≤ 1Presumably due toroughness and contaminants on the
n
ry
yrE
⎥⎥⎦
⎤
⎢⎢⎣
⎡+= 2,2, 1 εσ
σσ 0.0141
= 0.0316
,
2,rεcontaminants on thesurface.
y ⎦⎣
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Yu & Blanchard, J. Mater. Res. 11, 9 (1996)
⎟⎠⎞
⎜⎝⎛ −=
RaC Ra 4921.0845.2, λ( )
⎪
⎪⎨
⎧≤≤
=arb
ar-p
rpm for 1
23
2
2
⎠⎝ R( )⎪⎩
⎨≤≤ brC yRa 0 for , σ
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⎧
Yu & Blanchard, J. Mater. Res. 11, 9 (1996)
⎪⎪⎪⎪⎧
≤−
−<
−Ra
RaE
RaE y
y
14921.0845.2
)1(2
0 for )1(3
4 2
2
συπσυπ
⎪⎪⎪
⎪⎪⎪
⎨
∞<−
<⎟⎟⎟⎟⎞
⎜⎜⎜⎜⎛
⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛ −
=
RaE
Ra
a
RH
y
y
)1(2
1for 4921.0845.2
-14921.0845.2 2
2
2
συπσ
⎪⎪⎪⎪
⎩
−⎟⎟⎟
⎠⎜⎜⎜
⎝⎟⎟⎠
⎞⎜⎜⎝
⎛
−
⎟⎠
⎜⎝
Ra
RaER
y
4921.0845.2)1(
232
2 συπ
( ) ( ) ( )( )Ea
RaRaR
H
y
yyy
yy
2222242
2222222
021.520299.0
13459.011730.094.18 σ
σσυσυ
συσυ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+−
+−−−−−
=REa
H 22=
Apply P, measure h and S, assume E to get A, @ h = 20 nm (full contact), H = 1186.12 MPa
= 424 MPa, relative error = 55.3%yσ
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Field and Swain, J. Mater. Res. 10, 1 (1995)2
P ca'20hrh
rte hhh −=2e
rbh
hh +=
2
⎟⎠⎞
⎜⎝⎛
+−
=1412
252
nnc
PdP 3
32
⎟⎟⎠
⎞⎜⎜⎝
⎛=
s
t
PP
r
2)'( 8.2 caP
r πσ =
Rca
r 2.0=ε1−−
=r
hrhh ts
r
22' bb hRha −=ee h
PdhdP
23
=
Pt and ht
P and hPs and hs
Field and Swain (1993)
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Field and Swain, J. Mater. Res. 10, 1 (1995)
0.50 20 40 60 80 100Al 6061-T6
0.4Field & SwainP/P
(mN
) .
0.3
Sam
ple
peak load
0.1
0.2
oad
on S
0
0.1
0 20 40 60 80 100
Lo 50% unloaded
0 20 40 60 80 100Displacement Into Surface (nm)
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Field and Swain, J. Mater. Res. 10, 1 (1995)
• slope = Meyer’s index, m+ 2-0 6
Al 6061‐T6
•m + 2 = nwhere• n = 0.5491
il d 0 0931
-0.8
0.6
Y = ‐6.7719 + 2.5491X R = 0.99937nkεσ =
• n, tensile data = 0.093
•-1.2
-1
og (P
)
h = 100 nm ⎟⎟⎠
⎞⎜⎜⎝
⎛+−
=1412
252
nnc
• c = 0.893 → sink‐in-1.6
-1.4Lo ⎠⎝ +142 n
• The stiffness equation:
-2
-1.8
1 9 2 2 1 2 2 2 3 2 4 ASEr 2
π=1.9 2 2.1 2.2 2.3 2.4
Log (a')A2
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Field and Swain, J. Mater. Res. 10, 1 (1995)⎞⎛2
P ca'20hrh
rte hhh −=2e
rbh
hh +=
2
⎟⎠⎞
⎜⎝⎛
+−
=1412
252
nnc
PdP 3
32
⎟⎟⎠
⎞⎜⎜⎝
⎛=
s
t
PP
r ASEr 2
π=
2)'(caP
r πσ =
Rca
r 2.0=ε1−−
=r
hrhh ts
r
22' bb hRha −=ee h
PdhdPS
23
==
Pt and ht
P and hPs and hs
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FIELD AND SWAIN ‐ TABOR: TENSILE VS. IIT
8000 0.03 0.06 0.09 0.12 0.15 0.18
0.2 a(Es=72.59GPa)/R (-)
600
700
800
Pa) 135.02.651 εσ
εσ
=
= nkAl 6061‐T6
400
500
600
ess
(MP 093.00.432 εσ =
h = 100 nm
200
300
400
rue
Stre
0
100
200 Tensile dataF and S, (P
m via A(E
s=72.59GPa))/2.8
Tr
00 0.03 0.06 0.09 0.12 0.15 0.18
True Strain (-)
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FIELD AND SWAIN ‐ TABOR: TENSILE VS. IIT
8000 0.03 0.06 0.09 0.12 0.15 0.18
0.2 a(Es=72.59GPa)/R (-)
600
700
800
Pa)
Al 6061‐T6135.02.651 εσ
εσ
=
= nk
400
500
600
ess
(MP 093.00.432 εσ =
200
300
400
rue
Stre
h = 100 nm
0
100
200 Tensile dataF and S, (P
m via A(E
s=72.59GPa))/2.8
Tr
00 0.03 0.06 0.09 0.12 0.15 0.18
True Strain (-)
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P/P ‐ TABOR: TENSILE VS. IIT.
8000 0.03 0.06 0.09 0.12 0.15 0.18
0.2 a(Es=72.59GPa)/R (-)
600
700
800
Pa)
h = 17 nm εσ = nk
h = 100 nm
400
500
600
ess
(MP h 17 nm
093.0
189.0
04320.742εσ
εσ
εσ
=
=
= k
200
300
400
Tensile datarue
Stre 0.432 εσ =
0
100
200 Tensile dataP/P, (P
m via A(E
s=72.59GPa))/2.8
Tr .
00 0.03 0.06 0.09 0.12 0.15 0.18
True Strain (-)
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PREVIOUS OBSERVATIONS: ISE
Indenters:Spherical indentation of Iridium
Swadener et al., J. Mech. Phys. Solids 50 (2002)
• sapphire spheres,R = 69,122,318 mm
• diamond “sphere”,3.00 0.01 0.02 0.03 0.04
Effective strain, 0.2a/R
R = 14 mm• hardened steel ball,R = 1600 mm2.0
2.5
(GP
a)
14 μm
122
69 μm
Features:• H and 3s similarfor large spheres1.0
1.5
ardn
ess
( 122 μm1600 μm
318μm g p• H increases as R decreases
• increase in H with0 0
0.5
Ha
3σ ρG =1
bR
a/R parallels workhardening
0.00 0.05 0.1 0.15 0.2
a/R
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P/P ‐ TABOR: TENSILE VS. IIT.
6000 0.03 0.06 0.09 0.12 0.15 0.18
0.2 a(Es=72.59GPa)/R (-)
εσ = nk
500
600
Pa)
h = 100 nm
093.0
189.0
0.4322.561εσ
εσ
=
=h = 17 nm
300
400
ess
(MP 03 εσ
RE in k = 30%
RE in n = 103%
200
300
Tensile dataP/P, (P
m via A(E
s=72.59GPa))/3.7ru
e St
re
.⎟⎟⎠
⎞⎜⎜⎝
⎛=
n
yEkσ
σ
0
100Tr
Al 6061‐T6 MPa 8.180=∴
⎟⎠
⎜⎝
y
y
σ
σ
RE i 34%σ00 0.03 0.06 0.09 0.12 0.15 0.18
True Strain (-)RE in = ‐34%yσ
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SUMMARY AND CONCLUSIONS
1. Assumption of a perfect sphere.
b d h d ld b ll l d2. FEA based methods could not be experimentally implemented: proscribed depths < roughness and/or contaminants. Techniques are not ideally suited to investigating volumes of material that require small spheres.
3. Yu’s theoretical pressure distribution overestimated σy by 55%.
4. Field & Swain’s procedure overestimated the plastic flow curve by ~40%.
5. Work‐hardening from the mechanical polishing cannot fully account for these discrepancies.
6. One possible explanation: Indentation Size Effect.