E-Journal of Advanced Maintenance Vol.7-1 (2015) 96-101
Japan Society of Maintenology

Semi-Nondestructive Determination of Anisotropic Stress-Strain Curve
Using Asymmetric Indenters with Different Vertical Angles
Shigetaka OKANO1,*, Misa MIYABE1 and Masahito MOCHIZUKI1
1Graduate School of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565-0871, JapanABSTRACT
In this study, a new methodology is proposed for semi-nondestructive determination of anisotropic stress-strain
curves for steel by using asymmetric indenters with different vertical angles. The functional relation between theindentation load-depth curve and the representative stress for steel was derived from the π theorem, and then therelation was expressed as a function of the work-hardening coefficient and constant depending on the geometryof the indenter. With the proper selection of representative strain level varies according to the geometry ofindenter, this relation is directly dependent on the geometry of indenter. On the basis of π theorem and FE
solutions, the relation between the representative strain level and the geometry of the indenter was quantified.
Subsequently, specific function formulae indicating the relation between the indentation load-depth curve andthe representative stress for anisotropic steel were constructed with a focus on the effect of the degree ofanisotropy and the vertical angle of indenter. By using these formulae, the representative stresses for anisotropic
steel can be quantified at different representative strain levels and thus the anisotropic stress-strain curve canalso be obtained. The usefulness of developed procedure was evaluated through numerical simulation.
KEYWORDS

ARTICLE INFORMATION

indentation technique, asymmetric indenter, semi-nondestructive inspection,
anisotropic stress-strain curve, representative stress, vertical angle,
representative strain, π theorem, finite element analysis

Article history:
Received 1 November 2014
Accepted 27 March 2015

1.

Introduction

In recent years, it has become increasingly important to assess material degradation ofmechanical properties due to aging or serious accidents of in-service structures [1, 2]. It may also be
more important than ever for rational structural strength design to identify the mechanical properties,
such as the anisotropic stress-strain curve, of processed material —not raw material— in newly
manufactured structures. A nondestructive or semi-nondestructive procedure is desirable because it isapplicable to structural members in actual structures. The indentation technique has the advantages ofcompact apparatus and simple procedure in addition to semi-nondestructive procedure.
In this study, a new methodology for semi-nondestructive determination of the anisotropic
stress-strain curve for steel by using asymmetric indenters with different vertical angles is proposedon the basis of pi (π) theorem and finite element (FE) solutions. The usefulness of developed
procedure is discussed based on the results of numerical simulation to compare the estimated resultswith the input values for anisotropic stress-strain curve.

2.

π Theorem Solution for Functional Relation between Indentation
Load-Depth Curve and Representative Stress for SteelIt is difficult to understand the essential complexity of the relation between indentationload-depth curve and the mechanical properties of materials. Therefore, in this study, the functional
relation between the indentation load-depth curve and the representative stress for steel is derived
from the π theorem solution [3, 4]. The relation is expressed as the following equation (1) based ondimension analysis using the π theorem [5]. In other words, the coefficient of indentation*Corresponding author, E-mail: okano@mapse.eng.osaka-u.ac.jp

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

96

S. Okano, et al./ Semi-Nondestructive Determination of Anisotropic Stress –Strain Curve Using Asymmetric Indenters with Different Vertical Angles

load(L)-depth(h) curve (C ; L=Ch2) is defined by a function of Young’s modulus (E), representative
stress (σr), work hardening coefficient (n) and constant depending the geometry of indenter (φ).
C/σr = Π(E/σr, n, φ)(1)

3. FE Solution for Specific Formulae for Relation Between Indentation
Load-Depth Curve and Representative Stress for SteelSpecific formulae for the relation between the indentation load-depth curve and therepresentative stress for steel were constructed through FE analysis.
3.1. Numerical Model and Its Validation
Numerical simulations were performed by FE analysis using ABAQUS ver. 6.9-1. The FE modelfor the steel plate and the diamond indenter is a 1/4 symmetric model, as shown in Fig. 1(a). The
dimensions of the steel plate are 2 mm × 2 mm × 1 mm. The minimum and maximum size of theelements are 10 µm × 10 µm × 10 µm in each direction of the steel plate just below the indenter and100 µm × 100 µm × 100 µm approximately. Young’s modulus (E) and Poisson’s ratio (ν) for the steel
plate were set to 200GPa and 0.3, respectively. Similarly, those for the diamond indenter were set to1000GPa and 0.1 for an elastic body. The coefficient of dynamic friction between the indenter andsteel was set to 0.15 [6]. The symmetry condition was considered at the symmetry plane and norestraint was considered at the free edge. The vertical displacement was fully restrained at the bottomof the steel plate. The indentation load was calculated by using the reaction force on the superior
aspect of the diamond indenter.
Experimental verification and validation of the numerical model for the indentation test wasconducted by using high-strength structural steel, JIS SM490YB. The material property of the truestress-plastic strain relation was used with consideration of the strain rate effect on the plasticity
characteristics obtained experimentally from static and dynamic tensile tests. Experimentalindentation test was performed by using AIS2100 system made by Frontics, Co.. Dimensions ofspecimen are 50 mm × 50 mm × 12 mm. The penetration speed and maximum depth were set to 0.1
mm/min and 65 µm, respectively. Other experimental setup conditions are the same with those inprevious study [7]. As the results, it was confirmed that the simulation and experimental results of theindentation load-depth curve show good agreement, as shown in Fig. 1(b).

(a) Finite element model(b) Indentation load-depth curves obtained
Fig. 1. Numerical model for indentation test using Knoop indenter and its experimental validation

3.2. Quantification of Relation Between Representative Strain Level and Vertical Angle ofIndenterFirstly, the specific formula for the relation between the indentation load-depth curve and therepresentative stress for isotropic steel was constructed by using the Knoop indenter. The power law
relationship between the true stress and the plastic strain of steel was considered. The yield stress (σY)
and the work-hardening coefficient (n) were set from 400 MPa to 1000 MPa by 200 MPa and from97

S. Okano, et al./ Semi-Nondestructive Determination of Anisotropic Stress –Strain Curve Using Asymmetric Indenters with Different Vertical Angles

0.1 to 0.4 by 0.1, individually, as shown in Table 1. The coefficient of the indentation load-depth
curve estimated on the basis of the FE solution is also shown in Table 1.
Table 1 Mechanical properties used and coefficient of indentation load-depth curve estimatedσY(MPa)nC(GPa)4006008001000

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

0.1

0.2

0.3

0.4

110

137

170

210

151

180

214

258

186

217

253

296

217

249285328

The representative strain was estimated so that the unique relation between C/σr and E/σr wasachieved under all conditions of yield stress and work-hardening coefficient, as shown in Fig. 2. In thecase of the Knoop indenter, the representative strain was 0.029 and the relation between theindentation load-depth curve and the representative stress is expressed as the logarithmic function [5].
C/σr = –3.99 + 82.92 log[E/σr](2)

Indenters with varying vertical angles were labeled as indenter 1 to indenter 4. As shown in Table
2 and Fig. 3, the ratio of the long axis to the short axis remained the same value in the Knoop indenterand only the vertical angle was varied. The relation between the representative strain and the verticalangle (θ1) of each indenter was quantified, as shown in Fig. 4. The representative strain level became
larger with decreasing vertical angle (θ1) of the indenter. With the proper selection of vertical angles,
the representative stress for steel can be determined at different representative strain levels.

Table 2 Conditions of vertical angle of indenterθ1 (°)

θ2 (°)

Knoop indenter

172.5

130.0

indenter 1

169.3

112.7

indenter 2

176.2

153.8

indenter 3

177.5

162.3

indenter 4

178.1

166.7

Fig. 2. Unified relation between C/σr and E/σrdepending on representative strain

Fig. 3. Definition of vertical angle of indenter

Fig. 4. Quantitative relation between representativestrain and vertical angle of indenter

3.3. Quantification of Relation Between Degree of Anisotropy for Anisotropic Steel andIndentation Load-Depth Curve
Next, the effect of the degree of anisotropy (m = σx/σy ; σx>σy=σz) for anisotropic steel on theindentation load-depth curve was investigated. Table 3 shows the mechanical properties and the98

S. Okano, et al./ Semi-Nondestructive Determination of Anisotropic Stress –Strain Curve Using Asymmetric Indenters with Different Vertical Angles

degree of anisotropy used in the numerical analysis, and Fig. 5 shows the definition of the directional
relation between the anisotropic stress in plane and the asymmetric indenter. On the basis of aparametric analysis, the relation between the indentation load-depth curve and the degree ofanisotropy for anisotropic material was estimated, as shown in Fig. 6. The ratio, CA/CB, for thecoefficient of the indentation load-depth curve obtained from direction A to direction B decreases
approximately linearly with increasing degree of anisotropy. The relation is expressed by thefollowing equation. It doesn’t strictly agree with the average data only in range that the parameter m issmall, however it is more important to agree with the average data in that the parameter m is larger.
CA/CB = 1.31 – 0.31 m(3)Table 3 Mechanical properties and degree of anisotropy usedσY in σy and σz (MPa)nm

4000.1, 0.2, 0.3, 0.4
1.1, 1.3, 1.5

600
0.1, 0.2, 0.3, 0.4
1.1, 1.3, 1.5

8000.1, 0.2, 0.3, 0.4
1.1, 1.3, 1.5

1000
0.1, 0.2, 0.3, 0.4
1.1, 1.3, 1.5

Fig. 6. Effect of degree of anisotropy on CA/CAFig. 5. Defined directions for asymmetric indenter

3.4. Indentifying Specific Function Formulae for Relation Between Indentation Load-Depth
Curve and Representative Stress for Anisotropic Steel
Finally, the relation between the indentation load-depth curve and the representative stress in theanisotropic material was estimated on the basis of the FE solution. Here, the representative stresscorresponds to the lower stress component, σy. The relation between the coefficient of the indentationload-depth curve, CA, and the representative stress, σy, was expressed as
CA/σy = A + B log[E/σy](4)where the constants A and B depend on the vertical angle of the indenter and the degree of anisotropy.
This dependence was investigated through a parametric analysis, which showed that the effects of thevertical angle and the degree of anisotropy on the constants A and B, as shown in Figs. 7(a) and 7(b).

(a) Constant A(b) Constant B
Fig. 7. Effects of vertical angle of indenter and degree of anisotropy on constants A and B

99

S. Okano, et al./ Semi-Nondestructive Determination of Anisotropic Stress –Strain Curve Using Asymmetric Indenters with Different Vertical AnglesAccording to these figures, the effect of the vertical angle of indenter is much larger than that ofthe degree of anisotropy. Based on the results, it can be seen that A increases and B decreases withdecreasing vertical angle of the indenter. And then, the constants A and B were quantified in relation
to the vertical angle of indenter, as shown in Table 4.
Table 4 Estimated values of constant A and B in relation to vertical angle of indenter

AB

Knoop indenter
-1.99
80.49

indenter 1
19.03
26.08

indenter 2-208.95
372.40indenter 3-539.75
785.67indenter 4-1288.83
1414.04

4.

Estimating Anisotropic Stress-Strain Curve Using Developed Technique

4.1.

Development of Procedures for Estimating Anisotropic Stress-Strain Curve

In the developed technique using three asymmetric indenters with different vertical angles, theanisotropic stress-strain curve is determined according to the following procedures (also see Fig. 8):
Step 1 Estimate coefficients CA and CB from indentation load (L)-depth (h) curve (L=Ch2)
Step 2 Estimate degree of anisotropy m from Eq. (3) using coefficients CA and CB
Step 3 Estimate representative stress σy from Eq. (4) using constants A and B depending on angle θ1
Step 4 Estimate representative stress σx from the relation, σx = mσyFig. 8. Procedures for estimating anisotropic stress-strain curve

4.2.

An Example of Application of Developed Procedures

The other numerical simulation was performed to evaluate the accuracy of developed procedure
for anisotropic stress-strain curve determination. The solid lines in Fig. 9 show the input values foranisotropic stress-strain curve, in which the degree of anisotropy varies according to the strain level.
Three kinds of indenters, which were the Knoop indenter, indenter 1 and indenter 4, were used for
estimating the representative stress at three different representative strain levels. The indentationload-depth curves obtained by using these indenters in two different directions are shown in Fig. 10.
The indentation load became significantly larger with increasing vertical angle of the indenter and, ineither case, was influenced by the directional relation between the asymmetric indenter and theanisotropic stress-strain field. And then, the coefficients of indentation load-curve were estimated, asalso shown in Fig. 10. Through the procedures described above, the anisotropic representative stresses
were estimated at three different representative strain levels depending on the vertical angle of theindenter. The estimated results, which are also shown in Fig. 9, were in good agreement with the input
values.

100

S. Okano, et al./ Semi-Nondestructive Determination of Anisotropic Stress –Strain Curve Using Asymmetric Indenters with Different Vertical Angles

Fig. 9. Input and estimated values of anisotropic
stress-strain curve

5.

Fig. 10. Indentation load-depth curve obtained byusing asymmetric indenters

Conclusion

A new methodology was proposed for semi-nondestructive determination of anisotropic
stress-strain curve using asymmetric indenters with different vertical angles. The method was thenapplied through numerical simulation to show its usefulness. The conclusions reached in this study are
as follows.
(1) On the basis of the π theorem and FE solutions, the functional relation between the indentationload-depth curve and the representative stress, which is a function of the work-hardening
coefficient and the geometry of indenter, was derived and then specific formulae wereconstructed.
(2) The effect of the vertical angle of the indenter on the representative strain was quantified. With
the proper selection of asymmetric indenters with different vertical angles corresponding to acertain level of representative strain, the representative stress can be obtained at different
representative strain levels.
(3) The effect of the degree of anisotropy on the indentation load-depth curve was quantified. By
using the ratio of coefficients of the indentation load-depth curve obtained for anisotropic steel,the degree of anisotropy can be estimated.
(4) Constants in the relational equation between the indentation load-depth curve and therepresentative stress in anisotropic steel were quantified in relation to the vertical angle of theindenter. By using the coefficients of indentation load-depth curve, the anisotropic stresses foranisotropic steel can be estimated.
(5) A procedure for estimating the anisotropic stress-strain curve by using asymmetric indenters withdifferent vertical angles was developed. The developed procedure was then applied throughnumerical simulation to confirm the accuracy of anisotropic stress-strain curve determination.

References
[1]
[2]
[3]
[4]
[5]
[6]
[7]

Y. Saito, T. Shoji, H. Takahashi: “The Material Degradation of SUS 316 HTB Used for Boiler Superheater
during Service Operation”, Trans. Jpn. Soc. Mech. Eng. A, 52(1986) 473 165-169. (in Japanese)
Y. Iwabuchi, N. Chiba, T. Hatano: “Estimation of Long-Term Degradation of Cr-Mo-V Steel Castings”, J.
Soc. Mater. Sci., Japan, 35(1986) 390 280-286. (in Japanese)
M. Dao, N. Choccacoop, K. J. Van Vliet, T. A. Venkatesh, S. Suresh: “Computational Modeling of theForward and Reverse Problems in Instrumented Sharp Indentation”, Acta Mater., 49 (2001) 3899-3918.
J. L. Bucaille, S. Stauss, E. Felder, J. Michler: “Determination of Plastic Properties of Metals by
Instrumented Indentation using Different Sharp Indenters”, Acta Mater., 51 (2003) 1663-1678.
N. Ogasawara, N. Chiba, X. Chen: “Representative Strain of Indentation Analysis”, J. Mater. Res., 20(2005) 8 2225-2234.
S. Tanimura, K. Kaizu, K. Mimura, H. Igaki: “Measuring Method for Coefficients of Impulsive Friction
and its Application”, Trans. Jpn. Soc. Mech. Eng. A, 61(1995) 591 2456-2460. (in Japanese)
S. Okano, M. Mochizuki: “Proposing a New Semi-nondestructive Measurement of Non-equibiaxial
Residual Stress Field using Indentation Technique without Reference Value of Hardness under Non-StressState”, Trans. ASME, 80(2014) 820 SMM0355. (in Japanese)101