ISIJ International, Vol. 40 (2000), No. 4, pp. 356–363
Mechanical Behavior of Carbon Steels in the Temperature Range
of Mushy Zone
Dong Jin SEOL, Young Mok WON, Kyu Hwan OH, Yong Chang SHIN1) and Chang Hee YIM1)
School of Materials Science and Engineering and Research Institute of Advanced Materials, Seoul National University, San
56-1, Shinrim-dong, Kwanak-ku, Seoul 151-742 Korea.
1) Iron & Steel Making Research Team, Technical Research
Laboratories, POSCO, Pohang, P.O. Box 36, 1, Koedong-dong, Kyungbuk, 790-785 Korea.
(Received on September 2, 1999; accepted in final form on November 10, 1999 )
Tensile strength and ductility of carbon steels have been measured in the temperature range of mushy
zone by the in-situ melting tensile test technique with Gleeble system. The specimen was melted and
cooled to the test temperature before the tensile deformation in order to get the mechanical properties subject to the continuous casting process. During hot tensile test, a ceramic fiber tube was used to reduce the
radial temperature gradient in the heated specimen. Tensile strength of carbon steels in the temperature
range of mushy zone increased with decreasing test temperature, and was well described by the modified
yield criterion for porous metals. The measured zero strength temperature (ZST) and zero ductility temperature (ZDT) were related to the solid fractions evaluated by the numerical simulation of microsegregation developed by Ueshima et al. The characteristic solid fractions of ZST and ZDT which corresponded to 0.75 and
0.99, respectively, were well described by the prediction equation on ZST and ZDT at given steel compositions and cooling rates.
KEY WORDS: tensile strength; mushy zone; zero strength temperature (ZST); zero ductility temperature
(ZDT).
1.
phases in the solidification temperature range. They reported that the tensile strength of carbon steel was dependent
on the phase state and ZST and ZDT corresponding to the
solid fraction of about 0.6–0.8 with various carbon contents. Won et al.8) reported that the critical solid fractions at
ZST and ZDT corresponded to 0.75 and 0.99, respectively,
which were determined by the statistical assessment of experimentally measured data with the microsegregation
analysis at various carbon contents and cooling rates. To
determine ZST and ZDT in the temperature range of mushy
zone by high temperature tensile test, the precise control
and evaluation of temperatures corresponding to the mechanical properties are important.
The objective of present study is to measure the mechanical properties and characteristic temperatures of carbon
steels with various carbon contents during and after solidification. For this purpose, the high temperature tensile tests
have been conducted using Gleeble 1500 system with specimens covered with quartz and ceramic tubes.9) The measured mechanical properties were analyzed with relation to
the solid fractions evaluated by a microsegregation analysis.
The obtained characteristic temperatures of carbon steels
were compared with the calculated temperatures by Won et
al.8)
Introduction
Continuous casting process has been developed toward
higher casting speed and thinner strand thickness to increase productivity, which inevitably gives the high probability of cracking in the strand. To reduce cracking during
continuous casting, it has become more important to control the process condition and microstructure in optimal
manner. The process control for high quality steel products
necessarily needs the better understandings on the high
temperature deformation behavior of steels, because the internal and longitudinal surface cracks tend to occur in a
brittle temperature range by thermal contraction and mechanical deformation.1)
The mechanical properties of steels in the temperature
range of mushy zone can be characterized by the zero
strength temperature (ZST) and zero ductility temperature
(ZDT),1) which have been investigated by the high temperature tensile tests subject to the in-situ melting and solidifying thermal history.2–5) The ZDT has been reported to be
lower than ZST by the experimental measurements.4–6) Shin
et al.4) reported that the solid fraction at ZST corresponded
to the range of 0.6–0.7, while that at ZDT corresponded to
1.0. Nakagawa et al.5) reported that the solid fractions at
ZST and ZDT were 0.75 and 0.98, respectively, in high carbon steels. Yu et al.6) reported that ZST of low and medium
carbon steels on heating was different from that on cooling.
Mizukami et al.7) studied the tensile strength of d and g
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ISIJ International, Vol. 40 (2000), No. 4
Table 1.
Chemical composition of the specimens. (mass%)
Fig. 2. Schematic diagram of the thermal and deformation history for tensile test.
Fig. 1. Shape of the cylindrical tensile specimen.
2.
Experimental
was calculated to evaluate the mean temperature as the representative test temperature. The temperature distribution
over the cross section of specimen is given as a function of
radial coordinate, r, as follows;11)
2.1. Materials and Mechanical Testing
The chemical compositions of the specimens used for
cylindrical hot tensile tests are listed in Table 1. Samples L,
M and H denote the samples of low, medium and high carbon contents, respectively. Sample L was machined from
the as-cast slab. Samples M and H were from the as-cast
billet. Gleeble 1500 system was used for the tensile test.9)
The specimen was heated by electrical current along the
axial direction.
Figure 1 shows a schematic drawing of the tensile specimen covered with quartz and ceramic fiber tubes. The
length of the cylindrical specimen was 110 mm and the diameter was 10.6 mm. The longitudinal direction of the
specimen was normal to both the casting direction and the
columnar dendrite growth direction. The temperature of
specimen was controlled by R-type (Pt–Pt13%Rh) thermocouple welded to the midspan surface of the specimen, as
shown in Fig. 1. The quartz tube supported the melting
zone of about 10 mm in length. The ceramic fiber tube was
used to reduce the radial temperature gradient of the tensile
specimen by preventing the heat loss at specimen surface
mainly due to radiation.10) The ceramic fiber tube is weak
enough not to change the measured strength.
Figure 2 shows the experimental conditions schematically, the thermal history and strain rate of tensile test. The
specimen was rapidly heated up to T1 with the heating rate
of 15°C/s, held for 10 min at T1 for homogenization, then
heated to TL with the heating rate of 1°C/s. TL is the liquidus temperature of the sample. After being held at TL for
60 sec for melting, the specimen was cooled down to the
tensile test temperature with the cooling rate of 1°C/s. The
tensile load was applied during or after solidification at the
strain rate of 1022/s, under the vacuum atmosphere of about
531024 torr to reduce the radial thermal gradient.
T 5Ts1
(
)
g0 2 2
R 2r ..........................(1)
4k
where Ts is the temperature of specimen surface. g0, k and R
correspond to the amount of heat generation, thermal conductivity and specimen radius, respectively. The amount of
heat generation is given by
g 05
4k
R2
DT ................................(2)
where D T is the difference between the surface and central
axis temperatures. Assuming k as constant, g0 will be constant at given Ts. The mean temperature (Tm) will be described by
∫∫
∫∫
2p
R
Trdrdq
Tm 5
0
0
2p
0
R
5Ts1
1
DT .................(3)
2
rdrdq
0
Substituting Eq. (3) into Eq. (1), the radial position
–
which corresponds to the mean temperature is r5(1/√ 2)R.
9)
Figure 3 shows the measured temperatures at surface
and center axis with the calculated mean temperature by
Eq. (3). As the surface temperature increases, the difference
between surface and center axis temperature increases. We
took the mean temperature as the test temperature in this
study.
2.2. Temperature Analysis of Specimen
Considering the temperature gradient along the radial direction of specimen in spite of covering the specimen with
ceramic fiber tubes, the representative temperature, which
corresponds to the measured properties of strength and ductility, was taken as the averaged temperature over the cross
section of specimen in this study. The temperature distribution over the cross section of cylindrical tensile specimen
3.
Results
Figure 4 shows the measured tensile strength of samples
L, M and H as a function of temperature in the temperature
range of mushy zone. The ZST was determined as the lowest temperature of zero strength. As the test temperature decreases below critical temperature of ZST, tensile strength
of each sample increases. The determined ZSTs are 1 505,
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ISIJ International, Vol. 40 (2000), No. 4
Fig. 3. Relationship between measured temperatures at surface
and center axis and the calculated mean temperature from
Eq. (3).
Fig. 4. Change of measured tensile strength of samples L, M and
H with the temperature in the mushy zone.
Fig. 5. Measred displacement to fracture of samples L, M and H
as a function of test temperature.
1 455 and 1 404°C in samples L, M and H, respectively.
Figure 5 shows the measured displacement to fracture of
samples L, M and H as a function of test temperature. The
phases of each sample are also given in the figures. The
phases were calculated using the microsegregation analysis.
As the test temperature decreases, the displacement to fracture increases below the critical temperature, ZDT, which is
the lowest temperature of zero ductility. The measured
ZDTs of samples L, M and H are 1 475, 1 374 and 1 314°C,
respectively. Below ZDT, marked increases in ductility
were observed in sample L and H, but the increase in ductility of sample M is relatively small because of the higher
impurity content such as P and S. Those P and S lower the
ductility of carbon steels especially near the solidification
temperature.1,3,5) Samples M and H show the continuous increases in ductility, but the ductility of sample L shows
hump with decreasing test temperature at about 1 450°C.
Figure 6 shows the change of solid fraction obtained by the
microsegregation analysis together with the measured displacement to fracture in sample L as a function of temperature. The d /g transformation of sample L starts at the temperature of 1 486°C during solidification and ends at
1 438°C, after the complete solidification at 1 466°C. The
© 2000 ISIJ
Fig. 6. Changes of solid fraction and displacement to fracture of
sample L with temperature.
low ductility of sample L at the temperature range of
1 466–1 438°C is considered to be the effect of the d /g
transformation. The local volume contraction due to the d /g
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ISIJ International, Vol. 40 (2000), No. 4
Table 2.
Measured ZST and ZDT of sample L, M and H.
Fig. 7. Appearances of fractured specimens (sample H) at different test temperatures.
transformation at the brittle temperature range gives rise to
the strain in addition to the thermal strain and increases the
possibility of cracking.1) After the completion of d /g transformation, the ductility of sample L increases rapidly as
shown in Fig. 6.
The measured ZSTs and ZDTs are listed in Table 2, including measured ZDT under faster cooling rate of 10°C/s.
The effect of cooling rate on characteristic temperatures becomes larger at the final stage of solidification, i.e., around
ZDT rather than at the initial stage because of the severe
segregation of solute elements at the final stage of solidification.8) The increase in cooling rate reduced the ZDT, as
shown in the table, because the increased cooling rate results in the decrease in solidification finishing temperature
due to more severe segregation of solute elements. The
measured ZSTs and ZDTs decrease with the increase in
carbon content.
Figure 7 shows the appearances of fractured sample H at
various test temperatures. The ductility increases with the
decrease in test temperature. At the test temperature of
1 309°C being just below ZDT, low reduction of area of
about 7 % was obtained. At the test temperature of 1 299°C
and lower, high reduction of area of about 60–80 % was obtained. The reduction of area shown in Fig. 7 coincides well
with the displacement to fracture shown in Fig. 5.
Figure 8 shows the SEM images of fractured surface at
test temperatures near the solidification temperature. Fig.
8(a) shows the fracture surface of the sample L at the test
temperature of 1 490°C, which is between ZST and ZDT of
sample L. The elongated and fractured dendrite arms can
be seen. Figs. 8(b) and 8(c) show the fracture surfaces
of the sample H at the test temperatures of 1 369°C and
1 299°C, respectively. The test temperature of 1 369°C lies
between ZST and ZDT, and 1 299°C lies below ZDT of
sample H. At the test temperature of 1 369°C, many elongated and fractured dendrite arms are observed (Fig. 8(b)).
Figs. 8(a) and 8(b) indicate that the carbon steels have
Fig. 8. SEM images of the fracture surface at various test temperatures (a) 1 490°C sample L, (b) 1 369°C sample H
and (c) 1 299°C sample H.
strength in the temperature range between ZST and ZDT,
because dendrite arms combine each other.4,5) Ductile intergranular dimple fracture, which was observed at the test
temperature of 1 299°C (Fig. 8(c)), indicates the completion
of solidification.
4.
Discussion
4.1. Stress Calculation in Mushy Zone
In order to obtain an expression for the flow curves of
carbon steels at various strain rates and temperatures, the
following relation is used;12)
e˙e5A exp(2Q/RT )[sinh(b k)]1/m with s e5Ke en......(4)
where e e is the effective plastic strain, se is the effective
stress, K is the strength coefficient, n is the strain hardening
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ISIJ International, Vol. 40 (2000), No. 4
exponent, Q is the activation energy for deformation, R is
the gas constant, m is the strain rate sensitivity, and A and b
are material parameters. The Eq. (4) can give the flow
stresses of d -phase and g -phase of carbon steels when those
parameters of A, Q, b , m and n are determined for each
phase.
The critical strain (e c) for internal crack formation can be
expressed as follows, taking the effect of the brittle temperature range (D TB) and strain rate into account as discussed
in literature;8,13)
e c5
j
e˙ m*DTBn*
Fig. 9. Schematic diagram of dendritic solidification assuming
that the secondary dendrite arms have sine profiles.
...............................(5)
where j is constant, m* and n* are the strain rate exponent
and the brittle temperature range exponent on the critical
strain, respectively. Clyne et al.14) considered the mushy
zone of continuously cast steels to be divided into liquid
feeding zone and cracking zone. Cracks formed in the liquid feeding zone are refilled with the surrounding liquid,
whereas cracks formed in the cracking zone can not be refilled with the liquid because the dendrite arms are compacted enough to resist feeding of the liquid. They proposed the solid fraction at the boundary between the two
zones to be 0.9. Davies et al.15) used the same value. Kim
et al.1) proposed the brittle temperature range as ZDT#
D TB#LIT and used the solid fraction of 0.9 at LIT as Clyne
and others suggested. The LIT and ZDT were taken as the
temperatures at which the solid fractions become 0.90 and
0.99,8) respectively, in this study. The evaluated parameters
in Eq. (5) from the experimental data of carbon steels were
reported as j =0.02821, m*=0.3131 and n*=0.8638.13) The
critical strain decreases with the increase in the strain rate
and the brittle temperature range, depending on the cooling
rate and the content of solute elements. With the increase in
the cooling rate and the content of solute elements, especially phosphorus and sulfur, the brittle temperature range
expands largely due to the severe segregation of solute elements at the final stage of solidification which is influenced
by the cooling rate.8)
The fracture stress of solidifying shell was estimated by
the submerged split chill tension testing method16–18) at high
strain rate range of 1021–1/sec. The high strain rate was
chosen to minimize the shell growth during the testing.
However, the typical strain rates due to deformation in continuous casting process have been reported as the order of
1024–1023/sec for all steels.19) In order to investigate the accurate picture for internal cracking, the critical stress of
steels in the mushy zone is required to be calculated based
on the measured critical strain at the same order of strain
rate in continuous casting. The critical stress of d -phase
and g -phase for internal crack formation in the mushy zone
is calculated using the Eqs. (4) and (5) as follows.20)
s cd 5
5
 e˙
 Q 
e cnd
⋅ sinh21 
exp d  
bd
 RT  
 Ad
j nd
e˙ m*⋅nd DTBn*⋅nd
© 2000 ISIJ
5
n
 e˙
 Qg  
e cg
exp
⋅ sinh21 

bg
 Ag
 RT  
j ng
n*⋅n
e˙ m*⋅ng DTB g
mg
 e˙
 Qg  
⋅ sinh 
exp

 Ag
 RT  
21
mg
.........(7)
The critical stress for crack formation of the steel, s c,
can be obtained through the rule of mixture;13,20)
s c5h · s 0 with s 05[d fss cd 1g fss cg ] .............(8)
where h is the hardening parameter during solidification
and s 0 is the critical stress of solid steels. The increase in
critical stress in the temperature range of mushy zone is related to the geometrical hardening due to the increase in interdendritic contact area with increasing solid fraction as
well as the material hardening which is a function of temperature, composition and phase. Geometrical and material
hardenings are represented by h and s 0, respectively.
Figure 9 shows a schematic drawing of dendritic solidification during which the secondary dendrite arms contact
and combine each other. Langer et al. assumed the shape of
secondary dentrite as a function of periodic Fourier-cosine
transform.21,22) For simplifying the modeling of the geometrical hardening parameter with dendritic structure, the secondary dendrite arms were assumed to have sine profiles in
this study. As the solidification proceeds, the solid fraction
reaches a critical point, cfs, at which the growing dendrite
arms begin to contact each other. As the solid fraction increases above cfs, the contact area ratio, L/l between two
adjacent secondary dendrite arms increases, where L is the
contact length of two dendrite arms and l is the secondary
dendrite arm spacing, as indicated in the figure. The geometrical hardening parameter, h , which was taken as L/l in
this study, can be related to the solid fraction, fs, as described in Appendix;
p
md
 e˙
 Q 
⋅ sinh 
exp d  
 RT  
 Ad
21
s cg 5
∫
h
0
c
(12z ) sin pz dz 5
f s2c f s
12c f s
, where cf s # f s #1 .....(9)
where fs is the critical solid fraction corresponding to ZST.
As the solid fraction, fs, varies from cfs to 1, h varies from 0
to 1.
Figure 10 shows the calculated geometrical hardening
parameter during solidification, h , as a function of solid
fraction. The solid line indicates the calculated h from Eq.
md
.........(6)
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ISIJ International, Vol. 40 (2000), No. 4
Fig. 10. Change of hardening parameter during solidification
with solid fraction.
Table 3.
Parameters of d and g phases in Eq. (4).9)
(9) and the dashed line indicates the linear function of solid
fraction which was used by previous workers.1,13) The calculation from Eq. (9) shows a steep increase in geometrical
hardening parameter at around the solid fraction of cfs and
1.0, whereas the linear function assumes that the geometrical hardening parameter increases linearly from 0 to 1 as
the solid fraction increases from cfs to 1 as follows.
h5
f s2c f s
12c f s
, where cf s # f s #1 ...............(10)
To calculate the critical stress of steels in the temperature
range of mushy zone, Eqs. (6), (7) and (9) were substituted
to the Eq. (8). Reporting value for parameters of d and g
phases in Eqs. (6) and (7) are listed in Table 3. The activation energies for deformation of d -phase and g -phase are
216.9 and 373.4 kJ/mol, respectively, which are close to the
values for lattice self diffusion in d -phase of 251 kJ/mol23)
and in g -phase of 330–420 kJ/mol.24) The strain hardening
exponent of d -phase, nd , which corresponds to 0.0379, is
smaller than that of g -phase, ng , which corresponds to
0.210.
Figure 11 compares the measured tensile strength with
the calculated critical stress by Eq. (8) for samples L, M
and H as a function of solid fraction. The solid fraction of
the mushy zone as a function of temperature was calculated
using the numerical microsegregation analysis proposed by
Ueshima et al.25) Tensile strength increases above the critical solid fraction because dendrites begin to have network
structures and an increase in contact area of growing dendrite arms,4,5) as shown schematically in Fig. 9. The critical
solid fraction in each sample is about 0.75. As the solid
fraction approaches 1.0, the tensile strength increases rapidly because of the rapid increase in the interdendritic contact
area due to the final solidification of interdendritic liquid
film, which was formed by solute microsegregation. The
calculated stresses from Eq. (8) using the hardening parameter during solidification (h ), which is expressed as an in-
Fig. 11. Measured and calculated tensile strength of sample L,
M and H as a function of solid fraction.
tegral function related to the solid fraction (Eq. (9)), describes the measured tensile strengths better than the calculated data using linearly approximated h (Eq. (10)).
4.2. Calculation of the Characteristic Temperatures
Figure 12 shows the measured ZST and ZDT of samples
L, M and H together with the non-equilibrium pseudo binary Fe–C phase diagram, which was calculated from the microsegregation analysis. The thick solid lines represent the
non-equilibrium phase diagram and the thin solid lines represent the equilibrium binary Fe–C phase diagram. In the
case of sample L, at first the liq/d transformation occurs as
the temperature decreases from the liquidus temperature.
Then the d /g transformation takes place during and after
solidification. Sample M has the liq/d and d /g transformations just in the early stage of solidification, and then the
liq/g transformation occurs during most of the solidification
process. In the case of sample H, only liq/g transformation
occurs during solidification. The solid fraction at ZST of
each sample is about 0.75, and that at ZDT is about 0.99.
The calculated complete solidification temperatures are in
good agreement with measured ZDT. Recovery of ductility
is closely related to the completion of solidification of inter361
© 2000 ISIJ
ISIJ International, Vol. 40 (2000), No. 4
dendritic or grain boundary liquid film.26)
Based on the Clyne–Kurz microsegregation model27)
which takes the solute diffusion in solid into account, the
prediction equation for those characteristic temperatures of
ZST and ZDT was recently reported by Won et al.8) a
s functions of steel composition and cooling rate;

T 515362


tion equation, which assumed k as constant, was in reasonable agreement with the calculation by the Clyne–Kurz
model for various solute elements.
To calculate the characteristic temperatures of carbon
steels by Eq. (11), the corresponding solid fractions of ZST
and ZDT were used as 0.75 and 0.99, respectively, as was
reported by Won et al.8) and confirmed from the experimental results and microsegregation analysis in this work. The
measured characteristic temperatures of ZST and ZDT are
compared with the calculated values in Fig. 13, including
the measured data by previous workers.2,4,5) Those calculated characteristic temperatures with Eq. (11) well describe
the measured data of ZDT as well as those of ZST.
∑ f 9(C ) ⋅ [12 f (122W k )]
i
s
( k 21) /(122W k )
i
∑ f 9(C )567.51(wt%C)19.741(wt%Si)
i
i
13.292(wt%Mn)182.18(wt%P)1155.8(wt%S) .....(11)
W5a (12exp(21/a ))2
1
exp(21/2a ),
2
4.
Conclusions
High temperature tensile tests have been conducted
under in-situ melting and solidification thermal history
using cylindrical steel specimens covered with the ceramic
fiber tube, in order to measure the mechanical properties
and characteristic temperatures of carbon steels during and
after solidification at various carbon contents. The tensile
strength of carbon steels in the temperature range of mushy
zone was closely related to the formation of network structure of the growing dendrite arms and increased with decreasing test temperature because of the increase in solid
fraction. The recovery of ductility in carbon steels was observed around the complete solidification temperature and
the ductility increased with lowering test temperature. The
corresponding solid fractions of ZST and ZDT evaluated
using the microsegregation analysis were about 0.75 and
0.99, respectively. The experimentally measured tensile
strengths were well described by the modified yield criterion for porous metal with the integral form of geometrical
hardening parameter during solidification. Measured characteristic temperatures of carbon steels were in good agreement with those calculated values by the prediction equation on ZST and ZDT.
with a 533.7 · Ṫ 20.244
where f 9(Ci) is a function of the solute concentrations, fs is
the solid fraction, a and W are parameters expressing the
degree of back diffusion of solute element. k is 0.265, the
averaged equilibrium distribution coefficients for carbon of
kd /L and kg /L, because the equilibrium distribution coefficient in low and medium carbon concentrations, in which
the peritectic reaction occurs, can not be taken into account
in the prediction equation. Won and others reported that the
temperature–solid fraction relation calculated by the predic-
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ISIJ International, Vol. 40 (2000), No. 4
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y05
l9  c
2p 
c
x  ............(A-1)
 f s1(12 f s ) cos
2 
l 
where l 9 and l are the primary and secondary dendrite arm
spacing, respectively. In obtaining Eq. (A-1), the following
two conditions were used; y05l 9/2 when x50 and
(4/ll 9)∫l0 /2y0dx5cfs. When fs$cfs, the profile of secondary
dendrite arm can be obtained taking y5l 9/2 when x5L/2.
y5

l9  c
2L
2p  
1cos
x   ....(A-2)
 f s1(12c f s )12cos
2 
l
l  

The liquid area in Fig. 9 is shaded. The ratio of shaded area
to the total area means liquid fraction, which makes the following form.
4
ll 9
∫
l/2
L/2
 l9

c
 2 2 y  dx512 f s , where f s # f s #1 .....(A-3)


The integration of Eq. (A-3) after inserting Eq. (A-2) into
Eq. (A-3) gives the following result.

pL l
pL 
(12c f s )(l2L)cos
1 sin
5l (12 f s ) ........(A-4)
l
p
l 

Using the relation of L5lh , the Eq. (A-4) can be expressed as follows.
(12h ) cos ph 1
1
12 f
sin ph 5 c s ...........(A-5)
p
12 f s
Considering the integration by part, Eq. (A-5) can be expressed in another form as follows.
p
Appendix
∫
h
0
When the solid fraction reaches the critical value of ZST
363
(12z ) sin px dx 5
f s2c f s
12c f s
...............(A-6)
© 2000 ISIJ