Division II (include assigned division number from I to X)
USE OF AUTOMATED BALL INDENTATION TECHNIQUE TO
EVALUATE ROOM TEMPERATURE FRACTURE TOUGHNESS OF
NUCLEAR VESSEL STEELS
Guoxin Zhang1, Shang Wang1, Jirui Cheng1, Aiju Li2, Weiqiang Wang2*
1
Post graduate student, School of Mechanical Engineering, Shandong University, China
Engineering and Technology Research Centre for Special Equipment Safety of Shandong Province, China Research Centre of Safety Guarantee and Assessment to Special Equipment, Shandong University, China
2
Professor, School of Materials Science and Engineering, Shandong University, China
3
Professor, School of Mechanical Engineering, Shandong University, China
Engineering and Technology Research Centre for Special Equipment Safety of Shandong Province, China Research Centre of Safety Guarantee and Assessment to Special Equipment, Shandong University, China *Corresponding author ([email protected])
ABSTRACT
Determination of fracture toughness of pressure vessel steels using non-conventional techniques has been an active area of research for a long time. Among various small specimen or non-destructive techniques to determine fracture toughness of materials, the Automated Ball Indentation (ABI) technique has proved to be advantageous. ABI can be used to measure the fracture toughness of materials in service when a conventional fracture test cannot be performed. The present work highlights the applicability of ABI for evaluating the fracture toughness of SA516Gr70, and SA533B, which are two kinds of common steels used in nuclear vessels. At the same time, some standard destructive fracture tests were carried out for contrast. The results of the ABI tests show good agreement with those of the traditional tests. Meanwhile, some numerical simulations have been carried out by Finite Element Method (FEM) using ABAQUS software package. The FEM of the ABI simulation helps to improve the accuracy of the ABI method to determinate the fracture toughness by considering the extent of pile-up and sink-in on the edge of the indentation of the nuclear vessel materials.
INTRODUCTION
Nuclear energy has been developed and used by more and more countries in the world as a kind of non-pollution clean energy. However, because of the danger of the nuclear energy, it’s very important for the structural integrity assessment and safe use of the nuclear reactor pressure vessels. Fracture toughness, as the ability to resist crack extension, is a very important mechanical performance index for nuclear reactor pressure vessels in service. How to measure the fracture toughness non-destructively for nuclear reactor pressure vessels in service has been focused in recent years. Among various small sample or non-destructive techniques to determine fracture toughness of materials, the ABI technique has proved to be advantageous. ABI can be used to measure the fracture toughness of materials in service when a conventional fracture test cannot be performed.
method firstly and developed a complete set of testing equipment. Haggag (Haggag, 1993) applied this method to measure the yield strength and ultimate tensile strength of different materials and the obtained results agreed well with those from conventional tensile tests. Based on the above work, Byun et. al (Byun, Jin, Hong, 1998) first proposed an indentation energy to fracture (IEF) model that relates indentation deformation energy to fracture energy in order to evaluate fracture toughness from instrumented ball indentation. However, this model can be applied only to brittle materials or lower shelf energy level in the ductile–brittle transition temperature region of ductile material. After that, Haggag et.al improved this model and applied this model to evaluate the fracture toughness of ductile material. However, it can only be used for ferrite steel from their application. Lee et.al (Lee, Jang, Lee, 2006) suggested a new model(CIE) for estimating fracture toughness of ductile material based on continuum damage mechanics (CDM) principle. After that, many researchers used this model and got good results.
In the estimation of fracture toughness using ABI method by CIE model, the calculation result may be influenced by the pile-up or sink-in phenomenon. Thus, it is very important to modify it. There are many variables that maybe affect the height of the pile-up and sink-in, and many researchers (Biwa, Storakers, 1995, Hernot, Bartier, Bekouche, 2006) tried to analyze the problem, but so far there has not been a comprehensive conclusion and an empirical formula that can be used directly for calculations conveniently.
In this paper, some ABI tests and standard destructive fracture tests were carried out in order to evaluate the applicability of ABI for evaluating the fracture toughness of nuclear vessels of SA516Gr70 and SA533B which are the two kinds of common steels used in nuclear vessels. Meanwhile, the factors influencing the pile-up and sink-in were analyzed and an empirical formula about the height of pile-up and sink-in was given by some numerical simulations to reduce the calculation error of CIE model.
METHODOLOGY
Fracture Toughness
According to the Griffith theory (Griffith, 1921), for a crack with a length of 2a in an infinite plate under the remote uniform tensile stress, the fracture toughness, KJC, can be calculated by
C f
π
J
K
a
(1)Where
f is tensile stress, MPa, in far field when fracture is occurring. It can be obtained byf f
2
π
EW
a
(2)Where Wf is the fracture energy of unit crack surface. Combined Eq. (1) with Eq. (2), the fracture
toughness can be expressed as
C
2
fJ
K
EW
(3)Although the equation was derived from the infinite plate, but Irwin have proved that Eq. (3) is a universal relation.
To evaluate the fracture toughness KJC by the ABI method, the fracture energy Wf should be
determined only by the parameters that can be obtained from the ABI tests. Lee related the work in the process of indentation test and the fracture energy Wf, and they called the work when the depth of the
indenter reached a certain critical depth as critical indentation energy. It can be represented by
*
f 0 2
4
2 lim h d
h h
F
W h
d
(4)ball indenter and nonlinearity of the work-hardening of materials. That means F=sh. Based on above equations, we can obtain the fracture toughness by
C
π
ln
*J
Es
D
K
D h
(5)Critical Depth of the Indentation
The critical depth of the indentation cannot be determined directly by ABI tests, Lee et.al solved this problem by using the concepts of continuum damage mechanics (CDM). Kachanov defined the damage variable M in the literature (Kachanov, 1986), which can be expressed by Eq. (6).
D
S
M
S
(6)
Where S is the cross-sectional area of the sample; SD is the area of micro-defects such as holes and
inclusions in the cross section of the sample. The load along the vertical direction of the surface of material will lead to the local shear stress around the indentation. The shear stress can cause the nucleation, growth, and gather of the voids. As a result, the void volume fraction will increase with the increasing of load, and the elastic modulus will be reduced. It is assumed that the voids distribute evenly, and the radius of void is r, the interval distance is l. Then the damage variable and void volume fraction can be expressed by Eq. (7) and Eq. (8) respectively.
2 2
π
r
M
l
(7)3 3
4 π
3
r
f
l
(8)Combine Eq. (1) with Eq. (2), then we can get the relationship between M and f byEq. (9).
2 3 2 3 π 4 π 3
M f
(9)
Lemaitre (Lemaitre, 1985) associated the effective elastic modulus ED with the damage
variable M, and the relationship between the two variables can be expressed by
1
D
E
E
M
(10) Oliver et. Al (Oliver, Pharr, 1992)proposed that the degree of the material damage under the indenter increases with the increase of the depth of the indenter. The parameters obtained by the indentation test can be used to obtain the effective elastic modulus ED
2 D 2 c
1
2
1
π
i iE
A
E
L
(11)Where Ei and
i are the elastic modulus and Poisson’s ratio of the indenter respectively;
is the Poisson’s ratio of the material; Ac is the contact area between the indenter and material; L is the unloadingslope of the curve at each unloading stage in the load- depth curve obtained by the indentation test.
The effective elastic modulus corresponding to different depth can be obtained by Eq. (11). Then the critical indentation depth h* corresponding to ED* can be determined by extrapolation of ln ED–
ln h curve, where ED* can be determined by being brought f*=0.25(Andersson, 1985)into Eq. (9). Finally,
EXPRIMENTS
In order to verify the indentation model, the ABI test was carried out on the nuclear pressure vessel steels SA516Gr70 and SA533B using SSM-B4000TM system (ATC corporation, USA, as Figure 1 shows), whose load and displacement resolution is 0.01112N(0.025lbs) and 0.0254
μm
(0.001mil) respectively. The chemical compositions of the steels are listed in Table 1. The size of the samples is 10*10mm*40mm, which is large enough to avoid the size effect of samples. The maximum indentation depth is 0.1 mm and the number of loading-unloading circle is 8. The elastic modulus of material should be evaluated from unloading curve other than reloading curve to avoid the effect of back-stress relaxation. The experiment is controlled by displacement and the unloading ratio is around 40%. Since the beginning of the unloading curve is adequate to provide all parameters for the calculation of elastic modulus. The indenter used in this research is a tungsten carbide ball with Ei = 700GPa and
i= 0. 21.In order to verify the results of the ABI test, the conventional fracture test was also carried out on the same steels using INSTRON8803 testing machine. Single sample unloading compliance method was adopted according to GB/T 21143-2014.
Table 1 The chemical compositions of steels used in this study (wt%)
Materials C Si Mn P S Ni Mo Cr V
SA516Gr70 0.220 0.265 1.272 0.017 0.0039 0.188 0.01 0.186 0.0011
SA533B 0.195 0.282 1.409 0.013 0.0021 0.684 0.524 0.193 0.003
Figure 1
SSM-B4000TM systemRESULTES AND DISCUSSIONS
Fracture Toughness
clear that ED decreases with increasing depth due to the damage accumulation during indentation. The
critical depth of the indentation h*can be obtained as the depth corresponding to the critical elastic modulus ED*which can be determined by bringing f*=0.25 into Eq. (9). Then the fracture toughness can
be obtained by Eq. (5), as listed in Table 2.
0.00 0.02 0.04 0.06 0.08 0.10 0 100 200 300 400 500 600
SA516Gr70 test point 1 SA516Gr70 test point 2 SA516Gr70 test point 3 SA533B test point 1 SA533B test point 2 SA533B test point 3
Depth/mm
Lo ad / N SA533B SA516Gr70
3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.60 4.65 4.70 4.75 4.80 4.85 ln ( ED ) /GPa
ln(h)/m
ln(ED)=-0.16ln(h)+5.4 R2=0.99
ln(h)
ln(E
D
)
Figure 2 The load-depth curves in ABI tests Figure 3 Relationship between lnh and lnED
The JR resistance curves from conventional destructive fracture test are showed in Fig.4. The
0.2mm offset line is decided from GB/T 21143-2014. KJC can be converted from JIC by Eq. (12). All the
results are summarized in Table 2.
IC C
=
21
JEJ
K
(12)
0.0 0.2 0.4 0.6 0.8 1.0 0 100 200 300 400 500 Data Points
JRResistance Curve
0.2mm Offset Line JR=1913(a-0.2)
JR=618a^0.97
JR
/N/m
m
a/mm
0.0 0.2 0.4 0.6 0.8
0 100 200 300 400 500 Data Points JRResistance Curve
0.2mm Offset Line
JR=477a^0.86
JR=2475(a-0.2)
JR
/N/m
m
a/mm
(a) SA516Gr70 (b) SA533B Figure 4 JR resistance curves
The fracture toughness KJC obtained from ABI tests and conventional fracture tests are listed in
Table 2 and the comparison between the two results are shown in Figure 5. The errors from ABI tests and conventional fracture tests are 2.9% and 22.1% for SA516Gr70 and SA533 respectively. These errors can be accepted in engineering. So we can see the ABI tests can be used to evaluated the fracture toughness of the nuclear pressure vessels.
Dimensionless analysis
The pile-up and sink-in phenomena in the ABI test are showed in Figure 6.
A dimensionless
parameter
c
2was defined to characterize the pile-up and sink-in
phenomena.We can express
c
2by
2 t
c h h
(13)
Where
h
tis the true contact depth considering the pile-up and sink-in
phenomena.
h
is the depth
without considering the pile-up and sink-in
phenomena. It is obvious that pile-up appears when
c
2is greater than 1 and the sink-in appears when
c
2is less than 1.
There
aremany
factorsthat may affect the phenomena of pile-up and sink-in, such as
the elastic modulus
E,
the strain hardening exponent
n
, the yield strain
0, the depth of the
indenter
h
, and so on. We can write
c
2as the function of these factors by Eq. (14) according to
dimension analysis theory.
Table 2 Test results
steel
Category
Test
Number
KJC from ABI Tests/MPam0.5 KJC from Fracture Tests/MPam 0.5
Test Value Average Value Test Value Average Value
SA516Gr70
1 226.7
229.6
—
223.1
2 229.3 234.5
3 232.9 211.6
SA533B
1 227.9
227.4
184.9
186.3
2 228.5 186.1
3 225.7 187.9
0 1 2 3 4
0 50 100 150 200 250 300
ABI test data points ABI test average
Fracture test average Fracture test averageKJC=223.1MPa·m0.5
ABI test averageKJC=229.6MPa·m0.5
KJ
C
/M
Pa
·
m
0.5
Test number
0 1 2 3 4
0 50 100 150 200 250 300
KJ
C
/M
Pa
·
m
0.5
ABI test data points ABI test average
Fracture test average
Test number ABI test averageKJC=227.4MPa·m0.5
Fracture test averageK
JC=186.3MPa·m 0.5
Figure 6 The Pile-up and Sink-in phenomena
2
0
, , ,
, , ,
c
f E
n h D
(14)Where
f
is a dimensionless function;
is the Poisson’s ratio and it can be considered as a
constant value of 0.3 for all steels;
D
is the diameter of the indenter;
is the friction coefficient
which can be as a constant and be ignored in this study. Thus the indentation dimensionless
parameter
c
2can be expressed as Eq. (15) according to Buckingham theorem.
2 ' 0
,
,
c
f
n
h D
(15)Where
f’
is another dimensionless function different from
f
. It is clear that the indentation
dimensionless parameter
c
2can only be involved with steel properties and depth of the
indentation and it is not associated with elastic modulus. In order to analyse their relationship
quantitatively, some simulations were carried out by ABAQUS
software package..
An axisymmetric 2D finite element model is made to computationally simulate the
ABI test. The ball indenter is modelled as a rigid body with the same parameter as specified in
the test study i.e. diameter is 0.76mm. The steel under experimentation is modelled as deformable
body i.e. diameter is 3.52mm and height is 1.76mm. Considering both loading and geometric
symmetries, we use the four node axisymmetric reduction element. The FE model consists of
about 18046 elements. The region surrounds the indenter has fine mesh to model the high stress
gradient and to obtain an accurate data. We also place contact surface at both material and
indenter surfaces i.e. surface to surface contact. Ball indenter is the master surface while the plate
is slave surface. The other parameters used are: small sliding, hard contact and no friction.
Axisymmetric boundary conditions are imposed on the nodes on the axisymmetric axis. The
indenter moves down to penetrate the steel with the bottom of the specimen fixed. Figure 7
shows the boundary conditions applied for the simulation and Figure 8 represents meshing of the
undeformed model. In this work, the displacement of the indenter tip is controlled. The material
parameters used in FEA are listed in Table.3.
Table.3 Parameters used in the FEA
Variables Values used in the FEA
Young's modulus/GPa 200
The depth of indenter/mm 0.09, 0.10, 0.11, 0.12, 0.135, 0.15 Strain hardening exponent n 0.05, 0.1, 0.15, 0.2, 0.3
Yield strain ε0 0.0005, 0.001, 0.002, 0.0025, 0.004, 0.005
Figure 8 The model used in the finite element analysis
Figure 8 to Figure 10 give the relationship between the dimensionless parameter c2 and strain hardening exponent n, yield strain ε0 and dimensionless depth h/D. As can be seen from the figures, c
2
approximately meets second order function relation with n and linear relationship with ε0 and h/D
respectively. So we can write c2 the expression by Eq. (16).
0.05 0.10 0.15 0.20 0.25 0.30 1.00
1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
0=0.0005
0=0.001
0=0.002
0=0.0025
0=0.004
0=0.005
D
im
e
nsionl
e
ss
pa
ram
e
te
r
c
2
Strain hardening exponent n
0.000 0.001 0.002 0.003 0.004 0.005 1.00
1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40
D
im
ensionl
ess
pa
ram
ete
r
c
2
Yield strain 0
n=0.05
n=0.1
n=0.15
n=0.2
n=0.3
0.12 0.14 0.16 0.18 0.20 1.21
1.22 1.23 1.24 1.25 1.26 1.27
0=0.0005
0=0.001
0=0.002
0=0.0025
0=0.004
0=0.005
D
im
ensionl
ess
pa
ram
ete
r
c
2
Dimensionless depth h/D
Figure 10 Variation in c2 with h/D for n=0.1
2 2
1 2 0
= 1+ 1 1 h
c n n
D
(16)
Where
,
1,
2,
and
are constants and their values were obtained by the finite element
simulations, then the Eq. (16) was expressed as the following Eq. (17).
2 2
0
=1.515 1 1.5537 2.3476 1 21.2072 1 0.0329 h
c n n
D
(17)
We can get
n, ε0and
h/D through the ABI tests, then c 2can be calculated through Eq. (17) . And ED can be calculated by Eq. (13) and Eq. (11).
CONCLUDING REMARKS
In this paper, an ABI test method proposed by Lee has been employed to evaluate
fracture toughness value of important nuclear steels like SA516Gr70 and SA533B. The
correlation of ABI tested results with conventional test results proved its feasibility. Moreover,
the pile-up and sink-in were simulated by FEM and a relationship function between the
dimensionless parameter
c
2and
strain hardening exponentn
,
yield strain ε0and dimensionless depth
h
/
D
has been proposed with the dimension analysis theory, which can contribute to getting the
fracture toughness by ABI test more accurate and conveniently.
ABI can be used to measure the fracture toughness of materials in service when a
conventional fracture test cannot be performed. ABI method is a very promising method for the
structural integrity assessment and safe use of the nuclear reactor pressure vessels in the future.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the support of National Key Research and
Development Project, research on the key technology of micro damage testing of material
properties and development of testing instrument, through grant number 2016YFF0203005, and
acknowledge the support of National Natural Science Foundation of China, the investigation of
the mechanism and universality of material mechanical proprieties tested by automated ball
indentation method, through grant number 51475269.
Andersson H. (1977). ―Analysis of a model for void growth and coalescence ahead of a moving crack tip,‖ Journal of the Mechanics and Physics of Solids, 25 217-233.
Biwa S, Storakers B. (1995). ―An analysis of fully plastic Brinell indentation,‖ Journal of the Mechanics & Physics of Solids, 43 1303-1333.
Byun T S, Jin W K, Hong J H. (1998). ―A theoretical model for determination of fracture toughness of reactor pressure vessel steels in the transition region from automated ball indentation test,‖ Journal of Nuclear Materials, 252187-194.
Griffith A A. (1921). ―The phenomena of rupture and flow in solids,‖ Philosophical transactions of the royal society of London, 221 163-198.
Haggag F M, Lucas G E. (1983). ―Determination of lüders strains and flow properties in steels from hardness/microhardness tests,‖ Metallurgical and Materials Transactions A, 14 1607-1613. Haggag F M. (1993). ―Application of flow properties microprobe to evaluate gradients in weldment
properties,‖ International Trends in Welding Sciences and Technology, 629-635.
Hernot X, Bartier O, Bekouche Y, et al.(2006). ―Influence of penetration depth and mechanical properties on contact radius determination for spherical indentation,‖ International Journal of Solids & Structures, 43 4136-4153.
Hert H. (1882). ―Über die Berührung fester elastischer Körper,‖ Journal für die reine und angewandte Mathematik, 92 156-171.
Kachanov L M. (1986). ―Introduction to continuum damage mechanics,‖ Springer Science & Business Media.
Lee J S, Jang J L, Lee B W, et al. (2006).‖ An instrumented indentation technique for estimating fracture toughness of ductile materials: A critical indentation energy model based on continuum damage mechanics,‖ Acta Materialia, 54 1101-1109.
Lemaitre J. (1985). ―A continuous damage mechanics model for ductile fracture,‖ Journal of Engineering
Materials and Technology, 107 83-89.
Meyer E. (1908). ―Contribution to the knowledge of hardness and hardness testing,‖ Zeitschrift Des Vereines Deutscher Ingenieure, 52 740-835.
Oliver W C, Pharr G M. (1992). ―An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments,‖ Journal of Materials Research, 7 1564-1583.