Journal of Surface Engineered Materials and Advanced Technology, 2013, 3, 262-268 Published Online October 2013 (
Estimation of the Yield Stress of Stainless Steel from the
Vickers Hardness Taking Account of the Residual Stress
Osamu Takakuwa, Yusuke Kawaragi, Hitoshi Soyama
Department of Nanomechanics, Graduate School of Engineering, Tohoku University, Sendai, Japan.
Received July 1st, 2013; revised July 31st, 2013; accepted August 25th, 2013
Copyright © 2013 Osamu Takakuwa et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, a method that uses the Vickers hardness to estimate the yield stress of a metallic material with taking ac-
count of residual stress is proposed. Although the yield stress of bulk metal can be evaluated by a tensile test, it cannot
be applied to local yield stress varied by surface modification methods, such as the peening technique which introduces
high compressive residual stress at the surface. Therefore, to evaluate the local yield stress employing a relatively easy
way, the Vickers hardness test was conducted in this paper. Since the Vickers hardness depends on both the residual
stress and the yield stress, the relationship between the residual stress and the Vickers hardness was experimentally
examined. It was concluded that the yield stress of the surface treated by several peening techniques can be estimated
from the Vickers hardness once this has been corrected for residual stress.
Keywords: Peening; Yield Stress; Hardness; Residual Stress
1. Introduction
Peening is one of the most effective surface modification
techniques for introducing compressive residual stress
and work hardening metallic materials [1]. The mecha-
nical properties improved using this technique help to in-
crease the fatigue strength of mechanical components
and the resistance of structural materials to stress corro-
sion cracking. It is important to evaluate material proper-
ties such as the yield stress and the residual stress in the
surface after peening.
Although the residual stress in a layer modified by
peening can be easily measured by X-ray diffraction me-
thods, evaluating the yield stress of the modified layer is
much more difficult. The average yield stress of the ma-
terial as a whole can be evaluated by a general tensile test,
i.e., from the stress-strain curve. However, the local yield
stress altered by peening cannot be evaluated by this test
since this is predominantly affected by the base material
and less so by the modified layer.
Recently, indentation tests to evaluate the mechanical
properties in local areas of a material, have come to the
fore and several studies have been conducted on this
topic, and some approximate equations for evaluating the
yield stress from the load-displacement curve obtained
by the indentation test have been proposed [2-5]. Dao
et al. proposed a method to estimate the yield stress and
work hardening constant by an indentation test using a
conical indenter [2], which is based on the concept of
representative strain introduced by Tabor [6]. The repre-
sentative strain represents a strain field under the in-
denter during the indentation test. This concept made it
possible to evaluate the local mechanical properties. If
the conical indenter is used, two conical indenters with
different half apex angle at least would be needed. It is
because that two different representative strains need to be
obtained so as to determine the two unknown mechanical
properties, e.g., the yield stress and work hardening con-
stant. Also, Yan et al. proposed a method using a plural in-
denter based on a reverse analysis to evaluate these pa-
rameters on the residual stress field [7]. To make it easier,
a methodology employing a spherical indenter has been
proposed by Nishikawa et al. By the methodology, the
yield stress at the surface where residual stress exists can
be evaluated by a combination of an indentation test and
inverse analysis with response surface methodology us-
ing finite element analysis (FEA) [8]. However, this me-
thod is somewhat complex since much updating of the
response surface needs to be done for an accurate estima-
tion. For more practical use, a rapid and quantitative me-
thod needs to be developed. This study focuses on the rela
Copyright © 2013 SciRes. JSEMAT
Estimation of the Yield Stress of Stainless Steel from the Vickers Hardness Taking Account of the Residual Stress 263
tion between the yield stress, residual stress and the Vick-
ers hardness so as to rapidly and simply evaluate the lo-
cal yield stress of high compressive residual stress field
including treated materials by peening.
A simple method based on the Vickers hardness can be
used to estimate the yield stress. The relationship be-
tween the Vickers hardness and the yield stress has been
investigated [9,10]. Tabor [9] showed that the Vickers
hardness of several metals is proportional to the yield
stress, with a proportionality constant of approximately 3.
In the case of austenitic stainless steel, Busby et al. [10]
reported that the yield stress and Vickers hardness follow-
ed a linear relationship of the form Equation (1):
 (1)
where, ∆σy and HV, are the changes in the yield stress σy
and the Vickers hardness HV, respectively. K is a con-
stant determined experimentally, and has a value of 309
± 18 MPa GPa-1. The yield stress can be determined from
the measured value of the Vickers hardness using this li-
near relationship. However, the Vickers hardness is not
only affected by material properties such as the yield stress
and work hardening exponent but also the residual stress
[11,12]. The residual stress is easily induced by heat treat-
ment and/or mechanical processing. Therefore, in practi-
cal applications, the residual stress has to be taken into
account when evaluating the yield stress. In addition, the
relationship between yield stress and Vickers hardness is
difficult to apply to materials which have a residual stress
distribution induced by surface modification such as pee-
In this study, a simple and straight-forward method is
proposed, using the Vickers hardness test to estimate the
yield stress of JIS SUS316L stainless steel, in which com-
pressive residual stress is extant. First, in order to estab-
lish the pure relationship between the equibiaxial com-
pressive stress and the Vickers hardness, the Vickers hard-
ness tests were performed on specimens having several
value of the equibiaxial compressive stress. It was meas-
ured by X-ray diffraction method employing sin2ψ me-
thod. Second, in order to make sure that the regression co-
efficient obtained from the relationship between the com-
pressive stress and the hardness was independent of the
yield stress value, and simple finite element analysis was
done. Finally, the Vickers hardness tests and the indenta-
tion tests employing the combination of inverse analysis
with response surface methodology [8] were performed on
specimens treated by several peening techniques. Then the
relationship between the Vickers hardness and the yield
stress irrespective of the compressive residual stress was
determined. Therefore, the proposed method can be used
to determine the yield stress from the Vickers hardness.
2. Experimental Apparatus and Procedures
2.1. Evaluation of Vickers Hardness Varied by
Compressive Residual Stress
The material under test was austenitic stainless steel (Ja-
panese Industrial Standards JIS SUS316L). The geome-
try of each specimen was square, 35 mm on each side and
4 mm thick. In order to introduce and control the equibi-
axial compressive stress on the fronts of the specimens, a
cavitating jet was applied to the backs as shown in Fig-
ure 1 [13]. When the back side is exposed to the cavitat-
ing jet, plastic strain occurs due to impacts produced by
cavitation bubble collapsing. Then an equibiaxial elastic
reactive stress is generated against plastic deformation at
the back side. The reactive stress corresponds to equibia-
xial compressive stress induced by peening technique. So,
at the back side, the compressive residual stress is intro-
duced with increase in the yield stress due to work hard-
ening caused by the jet. If the thickness of the specimen
is certainly thin, curvature would be generated because of
lower resistance to the deformation due to the plastic strain
caused at the back side. Therefore, an equibiaxial com-
pressive stress on the front side is introduced by stretch-
ing the specimen without any work hardening. In this case,
neither plastic deformation nor a change in the crystalline
structure occurs on the front side of specimens, since the
cavitating jet is applied only to the back side. Therefore,
using this method, different compressive stresses can be
induced on the front sides of the specimens, while the
other material properties, such as Young’s modulus and
the yield stress, are the same in each case. The equibiax-
ial elastic-compressive stress induced by this method de-
pends on the equibiaxial curvature produced by the jet.
The equibiaxial curvature can be controlled by the expo-
sure time, i.e., amount of the treatment, and the thickness
of the specimen, i.e., the resistance to the bending. By
varying the exposure time to the cavitating jet, equibiaxi-
alelastic-compressive stress on the front side can be con-
trolled. A cavitating jet is generated when high-speed
Figure 1. Schematic illustration of cavitating jet apparatus
to introduce equibiaxial elastic compre ssive str ess.
Copyright © 2013 SciRes. JSEMAT
Estimation of the Yield Stress of Stainless Steel from the Vickers Hardness Taking Account of the Residual Stress
water is injected into water. To use a strong cavitating jet,
the pressure at which the high-speed water jet was inject-
ed, i.e., the upstream pressure, and the pressure of the
water filled in tank, i.e., the downstream pressure, were
set to 30 MPa and 0.42 MPa, respectively [14]. The noz-
zle diameter for the high-speed water jet, d, was 2 mm,
and the standoff distance between the nozzle throat and
the specimen, s, was 85 mm. Exposure times of 10, 20,
40, 80, 160, and 320 sec. were used. The residual stress
was measured in orthogonal directions at the centers on
the front sides of the specimens. It was confirmed that
the compressive residual stress introduced in the speci-
mens was almost equibiaxial [13].
Both the equibiaxial elastic stress,
ee, and residual
stress induced by peening,
r, were evaluated by the
method using an X-ray diffraction system MSF-
3M (Rigaku Corporation). The X-ray tube was a Cr tube
operated at 30 kV and 8 mA. X-rays from the Kβ peak
were used. The angle of the soller slit was 1 degree and
the width was 4 mm. The diffractive plane was the (3 1 1)
plane of γ-Fe. The reference diffractive angle 2
0 was
148.52 deg and the diffractive angle 2θ ranged from 143
deg to 153 deg in increments of 0.2 deg with 8 s intervals.
The stress factor for the X-ray diffraction measurements
was 368.93 MPa/deg. Vickers hardness tests were con-
ducted five times at the center of each specimen using a
micro hardness tester HMV-1 (Shimadzu Corporation).
The maximum load was set to Pmax = 1.961 N.
The yield stress of SUS316L used in the above expe-
riments was 300 MPa with 0.2% proof stress. In order to
confirm that the regression coefficient of hardness as a
function of residual stress does not depend on the varia-
tion in yield stress, it is necessary to investigate the rela-
tionship between the residual stress and the Vickers hard-
ness at various yield stresses. So, finite element analysis
FEA was done to just demonstrate that the relationship
between hardness and residual stress does not depend on
the value of the yield stress. The hardness at four values
of yield stress, varied by rolling rates of 0 (not-treated),
10 (Condition 1), 30 (Condition 2) and 50% (Condition
3), was obtained from FEA using the stress-strain curve.
The stress-strain curves of those specimens were measur-
ed by a tensile test using a precision universal tester AG-
I 50 kN (Shimadzu Corporation). The tensile rate was 1
mm/min. Thus, the relationship at various yield stresses
can be obtained by varying the residual stress. A finite
element model for the indentation test was constructed to
obtain the relationship between the residual stress and the
hardness of a specimen with a yield stress other than 300
MPa. Based on this model, elastic-plastic incremental
analyses using the commercial finite element code MSC.
Marc was conducted. Figure 2 illustrates the axisymmet-
ric finite element model used in the analyses. The inden-
ter was modeled as a rigid cone with a half-included tip
Figure 2. Finite element model for indentation test with a
conical indenter.
, of 70.3 deg, so as to conform the ratio of the
contact area to depth with a Vickers indenter. The speci-
men was modeled as a homogeneous isotropic elastic-
plastic material. The size of the specimen for the model
was 1 mm radius × 1 mm depth, which was sufficiently
large to exclude boundary effects. The indentation load
was applied through analysis of the contact between the
indenter and the specimen. Fixed boundary conditions
were applied to the bottom of the specimen, as shown in
Figure 2. Elastic-plastic analysis using an updated La-
grange configuration was conducted based on J2 flow
theory with isotropic hardening. Poisson’s ratio was set
= 0.3, and Young’s modulus and the work hardening
curve were measured by a uniaxial tensile test.
The hardness was calculated from the load, displace-
ment curve obtained by FEA. The hardness, H, is given
as follows:
where P and A denote the indentation load and the con-
tact area of the hardness impression, respectively. The
contact area, A, is a function of the contact depth, hc, and
is given by the following:
24.5 c
h (3)
where hc is given by the following Equation:
hh S
 (4)
where Pmax, h
, and S denote the peak load, the
maximum indentation depth, the geometric constant and
the contact stiffness, respectively. In the case of the co-
nical indenter,
= 0.726 [15]. The FEA was done to de-
monstrate that the relationship between hardness and re-
sidual stress does not depend on the value of the yield
Copyright © 2013 SciRes. JSEMAT
Estimation of the Yield Stress of Stainless Steel from the Vickers Hardness Taking Account of the Residual Stress 265
2.2. Evaluation of Vickers Hardness and the
Yield Stress Varied by Several Surface
In order to investigate the variation in Vickers hardness
with respect to the variation in yield stress, Vickers hard-
ness tests were conducted on several specimens with va-
rious yield stresses. The yield stress of these specimens
was varied by peening and polishing, according to the
conditions described in Tab le 1. In the cavitation peened
specimens, processing time per unit length, tp, was
changed to vary amount of the treatment. In the laser
peened specimens, laser energy, E, was also changed.
The yield stress of not-treated specimen (SR heat treat-
ment) and rolling rate of 50% specimen were 0.2% proof
stress determined by the stress-strain curve. The yield
stresses of polished, cavitation peened (CP) and laser
peened (LP) specimens were estimated by inverse analy-
sis using indentation by a spherical indenter [8], since the
tensile test cannot be used to evaluate the yield stress
distributed in the sub-surface. The inverse analysis pro-
cedure used to estimate the yield stress is described be-
low. This method has already constructed in the past re-
port [8]. The response surface relating the material para-
meters to the indentation load-displacement curves was
obtained using FEA of the indentation. Young’s modulus,
E, yield stress,
y, and the work hardening exponent, n,
were defined as material parameters. The response sur-
face enables the indentation curve of a material with arbi-
trary parameters to be estimated. Using inverse analysis
based on a genetic algorithm, the material parameters
were obtained by minimizing the error between the ap-
proximate values calculated by the response surface and
the experimental values obtained by the indentation tests.
Table 1. Treatment conditions for the specimen.
Evaluated by a tensile test Evaluated by inverse analysis
1 Not-treated 3 Polished
2 Rolling rate 50% 4
Cavitation peening using
Cavitating jet in water [16]
tp = 1 s/mm
Cavitation peening using
Cavitating jet in water [16]
tp = 4 s/mm
Cavitation peening using
Cavitating jet in water [16]
tp = 16 s/mm
Cavitation peening using
Cavitating jet in air [17]
tp = 20 s/mm
Laser peening conducted
by Cincinnati E = 6 J
Laser peening conducted
by Cincinnati E = 10 J
Laser peening conducted
by Toshiba Co. Ltd. [18]
E = 20 mJ
Additionally FEA of the indentation was conducted using
the material parameters found. If the accuracy of the re-
sponse surface was sufficiently high, the FEA result cor-
responded to the experimental indentation result. If not,
the FEA result was added to the data set of the response
surface to improve its accuracy. This updating process
was repeated until the accuracy of the response surface
was sufficiently high. The effect of residual stress was
eliminated by estimating the response surface from the
indentation load-displacement curve of a specimen with
no residual stress. The validity of this method was veri-
fied through application to specimens whose yield stress
was known. It had already been confirmed that the yield
stress obtained by inverse analysis corresponded well to
the yield stress measured by the tensile test [8].
3. Results and Discussion
3.1. Relationship between Vickers Hardness and
the Compressive Stress
Figure 3 shows the Vickers hardness, HV, as a function
of the equibiaxial elastic stress,
ee, obtained by a Vick-
ers hardness test conducted on specimens with various
residual stress. The compressive stress is presented as ne-
gative. It is confirmed that the maximum the equibiaxial
elastic-compressive stress induced on the front sides of
the specimens whose back sides had been exposed to the
cavitating jet was 250 MPa, which is less than the yield
stress of 300 MPa of the not-treated specimen, i.e., with-
in the range of elastic stress. The measured Vickers hard-
ness increases linearly with increasing compressive stress.
The line was approximated by the weighted least squares
method. The regression coefficient obtained from Figure
3 is (8.4 ± 1.4) × 104. The effect on Vickers hardness
due to the compressive stress was obtained by multiply-
ing the compressive stress by this gradient. This result
can be applied to estimate the Vickers hardness of a ma-
terial under compressive residual stress. According to past
studies [11,12], the effect of tensile residual stress on the
Vickers hardness is greater than that of compressive re-
sidual stress. In the case of compressive residual stress,
Figure 3. Relationship between residual stress and Vickers
hardness of SUS316L.
Copyright © 2013 SciRes. JSEMAT
Estimation of the Yield Stress of Stainless Steel from the Vickers Hardness Taking Account of the Residual Stress
such as in a peened surface, the gradient of the regression
line obtained from the present study should be used.
Figure 4 shows the FEA results of indentation with a
conical indenter. The applied compressive stress is lower
than the elastic limit. The hardness increases proportion-
ally with increasing compressive stress. The gradients of
the regression lines of the not-treated specimen, and Con-
dition 1 (10% rolled, σy = 453 MPa), Condition 2 (30%
rolled, σy = 734 MPa) and Condition 3 (50% rolled, σy =
839 MPa) specimens are (6.54 ± 0.13) × 104, (6.17 ±
0.54) × 104, (6.54 ± 0.35) × 104 and (6.43 ± 0.64) ×
104, respectively. These values are slightly smaller than
the experimental data. The reason for the difference might
be due to the different shape of the indenter. The equibia-
xial elastic stress field affects indented area significantly.
The tensile stress increases the indented area and the com-
pressive stress does opposite. So, Suresh and Giannako-
poulos proposed a method to estimate the residual stress
by variation of the indented area due to that stress [19].
The compressive stress corresponds to a reactive force
against the indenter. Therefore the relation between the
hardness and compressive stress should be independent
from yield stress and the gradient keeps constant even
though yield stress changes.
Since there is no significant difference in regression
coefficient between each condition, the relationship be-
tween the compressive stress and the hardness of
SUS316L is nearly the same in each case, even though
the yield stress is different. So, the effect of compressive
residual stress on the Vickers hardness of a specimen
modified by peening can be easily taken into account.
3.2. Relationship between Vickers Hardness and
the Yield Stress
Figure 5 shows the yield stress,
y, obtained by a tensile
test or inverse analysis with a spherical indenter as a
function of Vickers hardness, HV. The residual stress, and
the Vickers hardness both before and after correction,
and the yield stress of the treated specimens were shown
in Table 2. The open symbols show the Vickers hardness
before correction for residual stress and the closed sym-
Figure 4. Relationship between residual stress and hardness
for various yield stresses from FEA.
Figure 5. Relationship between yield stress and Vickers hard-
ness of SUS316L.
bols show the Vickers hardness after correction for re-
sidual stress using the regression coefficient obtained
from Figure 3. The residual stress was varied from 0 to
500 MPa depending on the conditions of treatment. The
yield stress of these specimens was estimated by inverse
analysis. The absolute values of the residual stresses of
the not-treated and rolling rate of 50% specimens could
be ignored, and no correction was made to the Vickers
hardness of the specimen, i.e., specimen No. 1 and No. 2.
The relationship between the yield stress,
y, and the Vi-
ckers hardness corrected for the effect of residual stress,
HV’, was found to be given by the linear approximation,
y = (332 ± 18)·HV (218 ± 30).
y and HV’ are ex-
pressed in units of MPa and GPa, respectively. The value
for without the residual stress correction deviates from
the regression line obtained after the residual stress cor-
rection, along with increase in Vickers hardness as shown
in Figure 5. The yield stress becomes higher, the com-
pressive residual stress is also getting higher. Then, the
effect of it on the Vickers hardness becomes larger. There-
fore the residual stress correction is more important than
the case of which compressive residual stress is relatively
small. According to Busby et al. [10], the proportionally
coefficient between yield stress and Vickers hardness of
austenitic stainless steel is 309 ± 18 MPa/GPa. The value
obtained in the present experiment is reasonably close to
the value in the previous report. This proportionally co-
efficient did not consider residual stress. In order to con-
firm the relevance of the prosed method taking account
of residual stress, Figure 6 plots the yield stress obtained
from the proposed method as a function of that from the
ref. [10] in the case of with and without correlation of re-
sidual stress. From the result shown in Figure 6, the gra-
dients of these regression lines show 0.84 for without the
correction and 1.04 for with the correction. When the re-
sidual stress correction is performed, the equation propo-
sed in ref. [10] can also be effective in the peened surface.
It was verified that the present method can be used to
estimate the yield stress of SUS316L in which equibiax-
ial residual stress is present.
Copyright © 2013 SciRes. JSEMAT
Estimation of the Yield Stress of Stainless Steel from the Vickers Hardness Taking Account of the Residual Stress
Copyright © 2013 SciRes. JSEMAT
Table 2. Residual stress, Vickers hardness and yield stress var ie d by several treatments.
Specimen number Treatment Residual stress
r MPa
Vickers hardness
Vickers hardness
Yield Stress
y MPa
1 Not-treated (SR treatment) 7 ± 9 1.58 ± 0.03 - 300
2 Rolling rate 50% 10 ± 22 3.14 ± 0.07 - 839
3 Polishing 146 ± 30 1.52 ± 0.06 1.40 ± 0.11 239 ± 27
4 Cavitation peening in water tp = 1 s/mm 231 ± 16 2.01 ± 0.08 1.81 ± 0.12 407 ± 45
5 Cavitation peening in water tp = 4 s/mm 246 ± 18 2.12 ± 0.06 1.92 ± 0.11 455 ± 40
6 Cavitation peening in water tp = 16 s/mm 351 ± 25 2.48 ± 0.03 2.19 ± 0.12 609 ± 63
7 Cavitation peening in air tp = 20 s/mm 479 ± 25 2.49 ± 0.06 2.08 ± 0.14 513 ± 68
8 Laser peening E = 6 J 506 ± 7 3.32 ± 0.03 2.90 ± 0.12 708 ± 63
9 Laser peening E = 10 J 514 ± 7 3.40 ± 0.07 2.97 ± 0.14 761 ± 49
10 Laser peening E = 20 mJ 405 ± 40 2.97 ± 0.08 2.63 ± 0.15 608 ± 37
Figure 6. Comparison of yield stress obtained from the pre-
sent method with from ref. [10] with and without residual
stress correction.
4. Conclusions
In conclusion, a straight-forward method for estimating
the yield stress of SUS316L stainless steel taking account
of the compressive residual stress was proposed in the
present paper. The procedure for estimating the yield
stress is as follows:
1) Measure the hardness HV using a Vickers hardness
tester and the residual stress
r by an X-ray diffraction
2) Obtain the corrected Vickers hardness HV’ from HV
and the compressive residual stress using HV’ = HV + (8.4
± 1.4) × 104
3) Estimate the yield stress using
y = (332 ±18) HV
(218 ± 30).
5. Acknowledgements
This work was partly supported by the Japan Society for
the Promotion of Science Research under the Grant-in-
Aid for Scientific Research (B)24360040. The authors
are grateful to Professor Vijay K. Vasudevan, University
of Cincinnati, and Dr. Yuji Sano, Toshiba Corporation,
for providing their laser peening apparatus. Some of the
results in this work were obtained using supercomputing
resources at the Cyber-science Center, Tohoku Univer-
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