Sensitivity to experimental errors determines the reliability and usefulness of any experimental investigation. Thus, it is important to understand how various test techniques are affected by expected experimental errors. Here, a semi-analytical method based on the concept of condition number is explored for systematic investigation of the sensitivity of spherical indentation to experimental errors. The method is employed to investigate the reliability of various possible spherical indentation protocols, providing a ranking of the selected data reduction protocols from least to most sensitive to experimental errors. Explicit Monte Carlo sensitivity analysis is employed to provide further insight of selected protocol, supporting the ranking. The results suggest that the proposed method for estimating the sensitivity to experimental errors is a useful tool. Moreover, in the case of spherical indentation, the experimental errors must be very small to give reliable material properties.

References

References
1.
Van Vliet
,
K. J.
,
Prchlik
,
L.
, and
Smith
,
J. F.
,
2004
, “
Direct Measurement of Indentation Frame Compliance
,”
J. Mater. Res.
,
19
(
1
), pp.
325
331
.10.1557/jmr.2004.19.1.325
2.
Field
,
J. S.
, and
Swain
,
M. V.
,
1995
, “
Determining the Mechanical Properties of Small Volumes of Material From Submicrometer Spherical Indentations
,”
J. Mater. Res.
,
10
(
1
), pp.
101
112
.10.1557/JMR.1995.0101
3.
Kim
,
J.-Y.
,
Kang
,
S.-K.
,
Lee
,
J.-J.
,
Jang
,
J.-I.
,
Lee
,
Y.-H.
, and
Kwon
,
D.
,
2007
, “
Influence of Surface-Roughness on Indentation Size Effect
,”
Acta Mater.
,
55
(
10
), pp.
3555
3562
.10.1016/j.actamat.2007.02.006
4.
Huang
,
Y.
,
Zhang
,
F.
,
Hwang
,
K. C.
,
Nix
,
W. D.
,
Pharr
,
G. M.
, and
Feng
,
G.
,
2006
, “
A Model of Size Effects in Nano-Indentation
,”
J. Mech. Phys. Solids
,
54
(
8
), pp.
1668
1686
.10.1016/j.jmps.2006.02.002
5.
Higuchi
,
R.
,
Mochizuki
,
M.
, and
Toyoda
,
M.
,
2010
, “
A Method for Evaluating the Stress–Strain Relationship of Materials in the Microstructural Region Using a Triangular Pyramidal Indenter
,”
Weld. Int.
,
24
(
8
), pp.
611
619
.10.1080/09507116.2010.486242
6.
Feng
,
G.
, and
Ngan
,
A. H. W.
,
2002
, “
Effects of Creep and Thermal Drift on Modulus Measurement Using Depth-Sensing Indentation
,”
J. Mater. Res.
,
17
(
3
), pp.
660
668
.10.1557/JMR.2002.0094
7.
Ahn
,
J.-H.
, and
Kwon
,
D.
,
2001
, “
Derivation of Plastic Stress–Strain Relationship From Ball Indentations: Examination of Strain Definition and Pileup Effect
,”
J. Mater. Res.
,
16
(
11
), pp.
3170
3178
.10.1557/JMR.2001.0437
8.
Jiang
,
P.
,
Zhang
,
T.
,
Feng
,
Y.
,
Yang
,
R.
, and
Liang
,
N.
,
2009
, “
Determination of Plastic Properties by Instrumented Spherical Indentation: Expanding Cavity Model and Similarity Solution Approach
,”
J. Mater. Res.
,
24
(
3
), pp.
1045
1053
.10.1557/jmr.2009.0108
9.
Lan
,
H.
, and
Venkatesh
,
T. A.
,
2007
, “
Determination of the Elastic and Plastic Properties of Materials Through Instrumented Indentation With Reduced Sensitivity
,”
Acta Mater.
,
55
(
6
), pp.
2025
2041
.10.1016/j.actamat.2006.11.012
10.
Lan
,
H.
, and
Venkatesh
,
T. A.
,
2007
, “
On the Sensitivity Characteristics in the Determination of the Elastic and Plastic Properties of Materials Through Multiple Indentation
,”
J. Mater. Res.
,
22
(
4
), pp.
1043
1063
.10.1557/jmr.2007.0123
11.
Lan
,
H.
, and
Venkatesh
,
T. A.
,
2007
, “
On the Uniqueness and Sensitivity Issues in Determining the Elastic and Plastic Properties of Power-Law Hardening Materials Through Sharp and Spherical Indentation
,”
Philos. Mag.
,
87
(
30
), pp.
4671
4729
.10.1080/14786430701589343
12.
Phadikar
,
J. K.
,
Bogetti
,
T. A.
, and
Karlsson
,
A. M.
,
2013
, “
On the Uniqueness and Sensitivity of Indentation Testing of Isotropic Materials
,”
Int. J. Solids Struct.
,
50
(
20–21
), pp.
3242
3253
.10.1016/j.ijsolstr.2013.05.028
13.
Oliver
,
W. C.
, and
Pharr
,
G. M.
,
1992
, “
An Improved Technique for Determining Hardness and Elastic Modulus Using Load and Displacement Sensing Indentation Experiments
,”
J. Mater. Res.
,
7
(
6
), pp.
1564
1583
.10.1557/JMR.1992.1564
14.
Phadikar
,
J. K.
,
Bogetti
,
T. A.
,
Kaliakin
,
V. N.
, and
Karlsson
,
A. M.
,
2013
, “
Conical Indentation of a Viscoelastic Sphere
,”
ASME J. Eng. Mater. Technol.
,
135
(
4
), p.
041001
.10.1115/1.4024395
15.
Cheng
,
Y.-T.
, and
Cheng
,
C.-M.
,
2004
, “
Scaling, Dimensional Analysis, and Indentation Measurements
,”
Mater. Sci. Eng.: R: Reports
,
44
(
4–5
), pp.
91
149
.10.1016/j.mser.2004.05.001
16.
Phadikar
,
J. K.
,
Bogetti
,
T. A.
, and
Karlsson
,
A. M.
,
2012
, “
On Establishing Elastic-Plastic Properties of a Sphere by Indentation Testing
,”
Int. J. Solids Struct.
,
49
(
14
), pp.
1961
1972
.10.1016/j.ijsolstr.2012.04.001
17.
Dieter
,
G. E.
, and
Bacon
,
D.
,
1986
,
Mechanical Metallurgy
,
McGraw-Hill
,
New York
.
18.
Lubliner
,
J.
,
1990
,
Plasticity Theory
,
Macmillan
,
New York
.
19.
Cao
,
Y. P.
,
Qian
,
X. Q.
,
Lu
,
J.
, and
Yao
,
Z. H.
,
2005
, “
An Energy-Based Method to Extract Plastic Properties of Metal Materials From Conical Indentation Tests
,”
J. Mater. Res.
,
20
(
5
), pp.
1194
1206
.10.1557/JMR.2005.0147
20.
Dao
,
M.
,
Lim
,
C.
, and
Suresh
,
S.
,
2003
, “
Mechanics of the Human Red Blood Cell Deformed by Optical Tweezers
,”
J. Mech. Phys. Solids
,
51
(
11–12
), pp.
2259
2280
.10.1016/j.jmps.2003.09.019
21.
Ogasawara
,
N.
,
Chiba
,
N.
, and
Chen
,
X.
,
2009
, “
A Simple Framework of Spherical Indentation for Measuring Elastoplastic Properties
,”
Mech. Mater.
,
41
(
9
), pp.
1025
1033
.10.1016/j.mechmat.2009.04.010
22.
Yan
,
J.
,
Chen
,
X.
, and
Karlsson
,
A.
,
2007
, “
Determining Equi-Biaxial Residual Stress and Mechanical Properties From the Force-Displacement Curves of Conical Microindentation
,”
ASME J. Eng. Mater. Technol.
,
129
(
2
), pp.
200
206
.10.1115/1.2400280
23.
Yan
,
J.
,
Karlsson
,
A.
, and
Chen
,
X.
,
2007
, “
Determining Plastic Properties of a Material With Residual Stress by Using Conical Indentation
,”
Int. J. Solids Struct.
,
44
(
11–12
), pp.
3720
3737
.10.1016/j.ijsolstr.2006.10.017
24.
Buckingham
,
E.
,
1914
, “
On Physically Similar Systems; Illustrations of the Use of Dimensional Equations
,”
Phys. Rev.
,
4
(
4
), pp.
345
376
.10.1103/PhysRev.4.345
25.
Alkorta
,
J.
,
Martínez-Esnaola
,
J. M.
, and
Sevillano
,
J. G.
,
2005
, “
Absence of One-to-One Correspondence Between Elastoplastic Properties and Sharp-Indentation Load–Penetration Data
,”
J. Mater. Res.
,
20
(
2
), pp.
432
437
.10.1557/JMR.2005.0053
26.
Capehart
,
T. W.
, and
Cheng
,
Y.-T.
,
2003
, “
Determining Constitutive Models From Conical Indentation: Sensitivity Analysis
,”
J. Mater. Res.
,
18
(
4
), pp.
827
832
.10.1557/JMR.2003.0113
27.
Cheng
,
Y.-T.
, and
Cheng
,
C.-M.
,
1999
, “
Can Stress–Strain Relationships be Obtained From Indentation Curves Using Conical and Pyramidal Indenters?
,”
J. Mater. Res.
,
14
(
9
), pp.
3493
3496
.10.1557/JMR.1999.0472
28.
Liu
,
L.
,
Ogasawara
,
N.
,
Chiba
,
N.
, and
Chen
,
X.
,
2009
, “
Can Indentation Technique Measure Unique Elastoplastic Properties?
,”
J. Mater. Res
,
24
(
3
), pp.
784
800
.10.1557/jmr.2009.0100
29.
Tho
,
K. K.
,
Swaddiwudhipong
,
S.
,
Liu
,
Z. S.
,
Zeng
,
K.
, and
Hua
,
J.
,
2004
, “
Uniqueness of Reverse Analysis From Conical Indentation Tests
,”
J. Mater. Res.
,
19
(
8
), pp.
2498
2502
.10.1557/JMR.2004.0306
30.
Cao
,
Y.
,
Qian
,
X.
, and
Huber
,
N.
,
2007
, “
Spherical Indentation Into Elastoplastic Materials: Indentation-Response Based Definitions of the Representative Strain
,”
Mater. Sci. Eng.: A
,
454–455
, pp.
1
13
.10.1016/j.msea.2007.01.002
31.
Zhao
,
M.
,
Ogasawara
,
N.
,
Chiba
,
N.
, and
Chen
,
X.
,
2006
, “
A New Approach to Measure the Elastic-Plastic Properties of Bulk Materials Using Spherical Indentation
,”
Acta Mater.
,
54
(
1
), pp.
23
32
.10.1016/j.actamat.2005.08.020
32.
Cao
,
Y. P.
, and
Lu
,
J.
,
2004
, “
A New Method to Extract the Plastic Properties of Metal Materials From an Instrumented Spherical Indentation Loading Curve
,”
Acta Mater.
,
52
(
13
), pp.
4023
4032
.10.1016/j.actamat.2004.05.018
33.
Xu
,
B.
, and
Chen
,
X.
,
2010
, “
Determining Engineering Stress-Strain Curve Directly From the Load-Depth Curve of Spherical Indentation Test
,”
J. Mater. Res.
,
25
(
12
), pp.
2297
2307
.10.1557/jmr.2010.0310
34.
Kang
,
B. S. J.
,
Yao
,
Z.
, and
Barbero
,
E. J.
,
2006
, “
Post-Yielding Stress-Strain Determination Using Spherical Indentation
,”
Mech. Adv. Mater. Struct.
,
13
(
2
), pp.
129
138
.10.1080/15376490500448607
35.
Haušild
,
P.
,
Materna
,
A.
, and
Nohava
,
J.
,
2012
, “
On the Identification of Stress–Strain Relation by Instrumented Indentation With Spherical Indenter
,”
Mater. Des.
,
37
, pp.
373
378
.10.1016/j.matdes.2012.01.025
36.
Kang
,
S.-K.
,
Kim
,
Y.-C.
,
Kim
,
K.-H.
,
Kim
,
J.-Y.
, and
Kwon
,
D.
,
2013
, “
Extended Expanding Cavity Model for Measurement of Flow Properties Using Instrumented Spherical Indentation
,”
Int. J. Plasticity
,
49
, pp.
1
15
.10.1016/j.ijplas.2013.02.014
37.
Chen
,
X.
,
Ogasawara
,
N.
,
Zhao
,
M.
, and
Chiba
,
N.
,
2007
, “
On the Uniqueness of Measuring Elastoplastic Properties From Indentation: The Indistinguishable Mystical Materials
,”
J. Mech. Phys. Solids
,
55
(
8
), pp.
1618
1660
.10.1016/j.jmps.2007.01.010
38.
Cao
,
Y. P.
, and
Lu
,
J.
,
2004
, “
Depth-Sensing Instrumented Indentation With Dual Sharp Indenters: Stability Analysis and Corresponding Regularization Schemes
,”
Acta Mater.
,
52
(
5
), pp.
1143
1153
.10.1016/j.actamat.2003.11.001
39.
Chollacoop
,
N.
,
Dao
,
M.
, and
Suresh
,
S.
,
2003
, “
Depth-Sensing Instrumented Indentation With Dual Sharp Indenters
,”
Acta Mater.
,
51
(
13
), pp.
3713
3729
.10.1016/S1359-6454(03)00186-1
40.
Hyun
,
H. C.
,
Kim
,
M.
,
Lee
,
J. H.
, and
Lee
,
H.
,
2011
, “
A Dual Conical Indentation Technique Based on FEA Solutions for Property Evaluation
,”
Mech. Mater.
,
43
(
6
), pp.
313
331
.10.1016/j.mechmat.2011.03.003
41.
Le
,
M.-Q.
,
2008
, “
A Computational Study on the Instrumented Sharp Indentations With Dual Indenters
,”
Int. J. Solids Struct.
,
45
(
10
), pp.
2818
2835
.10.1016/j.ijsolstr.2007.12.022
42.
Swaddiwudhipong
,
S.
,
Tho
,
K. K.
,
Liu
,
Z. S.
, and
Zeng
,
K.
,
2005
, “
Material Characterization Based on Dual Indenters
,”
Int. J. Solids Struct.
,
42
(
1
), pp.
69
83
.10.1016/j.ijsolstr.2004.07.027
43.
3DS,
2009
,
abaqus, Version 6.9-2
, Dassault Systèmes, Waltham, MA.
44.
Bowden
,
F. P.
, and
Tabor
,
D.
,
2001
, “
Appendix: Some Typical Values of Friction
,”
The Friction and Lubrication of Solids
,
Oxford University Press
,
New York
.
45.
Datta
,
B. N.
,
2010
,
Numerical Linear Algebra and Applications
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
, Chap. 4.
46.
Wang
,
L.
,
Ganor
,
M.
, and
Rokhlin
,
S. I.
,
2005
, “
Inverse Scaling Functions in Nanoindentation With Sharp Indenters: Determination of Material Properties
,”
J. Mater. Res.
,
20
(
4
), pp.
987–1001
.10.1557/JMR.2005.01
47.
Xu
,
Z.-H.
, and
Li
,
X.
,
2006
, “
Sample Size Effect on Nanoindentation of Micro-/Nanostructures
,”
Acta Mater.
,
54
(
6
), pp.
1699
1703
.10.1016/j.actamat.2005.11.043
48.
Moussa
,
C.
,
Hernot
,
X.
,
Bartier
,
O.
,
Delattre
,
G.
, and
Mauvoisin
,
G.
,
2014
, “
Evaluation of the Tensile Properties of a Material Through Spherical Indentation: Definition of an Average Representative Strain and a Confidence Domain
,”
J. Mater. Sci.
,
49
(
2
), pp.
592
603
.10.1007/s10853-013-7739-1
49.
Phadikar
,
J. K.
,
Bogetti
,
T. A.
, and
Karlsson
,
A. M.
, “
On the Construction of Confidence Interval for Indentation Testing
” (in press).
50.
Higham
,
N. J.
,
1996
,
Accuracy and Stability of Numerical Algorithms
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
, Chap. 7.
51.
Rheinboldt
,
W. C.
,
1976
, “
On Measures of Ill-Conditioning for Nonlinear Equations
,”
Math. Comput.
,
30
(
133
), pp.
104
111
.10.1090/S0025-5718-1976-0400702-1
You do not currently have access to this content.