### Estimations of the True Stress and True Strain until Just before Fracture by

### the Stepwise Tensile Test and Bridgman Equation for Various Metals and Alloys

### N. Tsuchida

1### , T. Inoue

2### and K. Enami

31_{Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan}
2_{National Institute for Materials Science, Tsukuba 305-0047, Japan}

3_{TOPY Industries, Limited, Toyohashi 441-8510, Japan}

True stress (·)true strain (¾) relationships until just before fracture, i.e., the plastic deformation limit, were estimated by the stepwise tensile test and the Bridgman equation for various metals and alloys with different crystal structures. The estimated·¾relationships were different from the nominal stressstrain curves including the conventional tensile properties. In the relationships between the true stress (·pdl) and

true strain (¾pdl) at the plastic deformation limit, SUS304 and SUS329J4L indicated a better·pdl¾pdlbalance. On the other hand, SUS329J4L,

tempered martensite, and an ultraﬁne-grained steel showed superior results in the yield strength¾pdlbalance. The estimated·¾relationship for

the ultraﬁne-grained steel suggests that grain reﬁnement strengthening can improve·and¾up until the plastic deformation limit. The value of ¾pdlbecame larger with increasing the reduction in area and a decrease in the fracture stress. The products of·pdland¾pdlbecame larger with

increasing work-hardening rate at the plastic deformation limit. [doi:10.2320/matertrans.MD201112]

(Received July 28, 2011; Accepted September 5, 2011; Published December 25, 2011) Keywords: true stress, true strain, tensile test, plastic deformation limit, fracture

1. Introduction

Stressstrain relationships play an important role in the
discussion of plastic deformation behavior of various
materials, and a tensile test to obtain the stressstrain
relationship has been widely used as a representative
mechanical test.1,2) _{The mechanical properties of }

ultraﬁne-grained materials have also been investigated by using stress
strain curves.38)_{In the tensile tests, nominal stress}_{}_{nominal}

strain curves, which can be obtained by load and elongation,
are usually shown. Local and total elongations largely depend
on the gage length.911)_{On the other hand, true stress (}_{·}_{) and}

true strain (¾) are real indicators of mechanical behavior and are more important than the nominal stress and nominal strain in understanding the plastic deformation behavior up until fracture. However, it is difﬁcult to measure · and ¾ after necking in conventional tensile tests.1,12)

In tensile tests using round test specimens, the state of
stress changes from uniaxial tension to a complex triaxial
tension condition as the neck develops in local
deforma-tion.12,13) _{Some approximate solutions to estimate} _{·} _{in the}

neck of round bars have been presented.1,12) _{Bridgman}12,13)

studied large plastic ﬂow and fracture, and he proposed an
approximate expression to estimate· during local
deforma-tion after necking in order to investigate the effects of
hydrostatic pressure on local elongation from the standpoint
of fracture. Marshall and Shaw14)_{examined the equation for}_{·}

derived by Bridgman and found that it was applicable for low
alloyed steel and copper, and this equation is referred to in
various papers.12,15,16) Recently, Enami et al.1719)
investi-gated the·¾relationship after necking for low carbon steels
by using the Bridgman equation and a stepwise tensile
test for a smooth round specimen. As a result, · and the
necking ratio change were evaluated as a function of¾(up to
a value of approximately 1.0). The value of · usually
continues increasing during tensile deformation,1,2) _{but}

Enamiet al.17,18)_{reported that the estimated}_{·}_{decreased just}

before fracture. In this study, the point just before fracture, in

which · starts decreasing, is called the plastic deformation limit and is considered as the initiation point for fracture.12,13) The ·¾ relationship up to the plastic deformation limit can be estimated by combining the stepwise tensile test and the Bridgman equation.17,18)

Materials, such as steels, are deformed until large true
strains of more than 1.0 occur in plastic working
processes,10,20)_{and the deformation behavior at high strains}

is also of great importance for bulk nanostructured metals because of severe plastic deformation procedures.21,22)

However, there is little data for the ·¾ relationship up
until fracture in various materials.1214,20) _{Comparisons of}

the ·¾ relationships among materials and the effects of
strengthening mechanisms on the ·¾ relationship also have
not been reported previously. In studying bulk
nanostruc-tured metals, the effect of grain reﬁnement strengthening
on the ·¾ curve should be clariﬁed. In studies of the
mechanical properties of ultraﬁne-grained steels, not only
high strength but also good ductility2326) and bending
properties9) _{have been reported. These must be associated}

with the local deformation behavior. Procedures for
under-standing the deformation behavior until fracture, regardless
of the specimen size, are necessary because the local
elongation is dependent on the specimen size.911) _{In}

addition, arrangements of mechanical properties using · and ¾ at the plastic deformation limit are not observed because the yield strength, tensile strength, total elongation, and so on, are usually used.26,27)

The present paper aims at revealing estimated ·¾
relationships up until the plastic deformation limit in various
metals and alloys with different crystal structures. The
stepwise tensile test to obtain the ·¾relationship until just
before fracture was also conducted by using an
ultraﬁne-grained low-carbon steel with a ferrite grain size of less
than 2 µm.8) _{By using the experimental results, we try to}

summarize the tensile properties from the standpoint of the

·¾ relationship and discuss the differences with conven-tional tensile properties.

Special Issue on Advanced Materials Science in Bulk Nanostructured Metals

2. Experimental Procedures

In this study, various commercial materials with different
crystal structures were used, and most of them were subjected
to heat treatment, as shown in Table 1. Some data refer to
experimental results from a previous study,17,18) _{and the}

specimen ufg-FC is the ultraﬁne-grained ferrite-cementite
steel with an average ferrite grain size of 1.5 µm.8)_{By using}

these materials, static tensile tests and stepwise tensile tests
in order to obtain · and ¾ until just before fracture were
conducted.17)_{Round tensile test specimens with a gage length}

of 40 mm and gage diameter of 8 mm were prepared from the
materials. Static tensile tests were performed with an initial
strain rate of 5©10¹4s¹1 (the constant crosshead speed of
0.02 mm/s) at 296 K by using a gear-driven type Instron
machine. In the stepwise tensile test, the load (P), radius of
the neck section (a), and the radius of curvature of the neck
proﬁle (R) were measured during the temporary stops of the
tensile test.12,17)_{In the measurements of}_{R}_{, the contour of the}

necked region was approximated by the arc of a circle.14)_{The}

change of estimated·by the measurement deviation ofRwas within approximately 10 MPa.

3. Estimations of True Stress and True Strain during Tensile Deformation

True stress (·) and true strain (¾) are usually calculated by using the following equations:

· ¼sð1þeÞ ð1Þ

¾¼lnð1þeÞ ð2Þ

where s and e are the nominal stress and nominal strain, respectively. Equations (1) and (2) are valid only as long as the deformation is uniform. After necking starts, in the case of round specimens, the radius of the neck cross section or the cross-sectional area must be measured independently to calculate ¾as,

¾¼2 lna0

a ð3Þ

whereais the minimum radius of the neck cross section, and

a0is the initial radius of the cross section.12,13)On the other

hand, the state of stress at the center of a neck is not uniaxial
tension, and thus, a stress distribution across the neck
appears.12) Figure 1 shows a schematic illustration of the
stress distribution in the neck of a round tensile test
specimen. Here, ·_{xx},·_{yy}, ·_{zz} are the true principal stress in
the radial, tangential, and axial directions, respectively.
Bridgman12,13) assumed the following conditions to obtain
an approximate expression for · after necking in round test
specimens.

·xx¼·yy ð4Þ

·¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1

2½ð·zz·xxÞ

2_{þ ð}_{·}

xx·yyÞ2þ ð·yy·zzÞ2

r

ð5Þ

Table 1 Summary of materials in the present study.

Material Materials composition (mass%) Crystal

structure Heat treatment

Aluminum (Al) 0.9982Al

fcc

723 K©3.6 ks (FC)

Copper (Cu) 0.9999Cu 773 K©3.6 ks (FC)

SUS310S (310S) 25Cr19Ni0.02C1.1Mn0.3Si 1373 K©1.8 ks (WQ)

Ferrite-Pearlite (FP1)

0.16C0.4Si1.5Mn0.013P0.004S

bcc

1473 K©3.6 ks (FC)

Ferrite-Pearlite (FP2) as-received

Ultraﬁne-grained Ferrite-Cementite

(ufg-FC) references (8)

Tempered Martensite (QT1)

0.2C0.08Si0.96Mn0.008P0.007S

1153 K©3.6 ks (FC), 823 K©5.4 ks (WQ)

Tempered Martensite (QT2) 1153 K©3.6 ks (FC), 678 K©5.4 ks (WQ)

Tempered Martensite (QT3) 1153 K©3.6 ks (FC), 563 K©5.4 ks (WQ)

SUS430 (430) 16Cr0.24Ni0.032C0.39Mn0.22Si 1123 K©3.6 ks (FC)

Titanium (Ti) Ti0.05Fe0.1O (JIS 2nd. grage) hcp 1123 K©3.6 ks (AC)

SUS329J1 (329J1) 24Cr5.5Ni0.03C0.8Mn0.6Si1.3Mo

fcc+bcc

as-received

SUS329J4L (329J4L) 25Cr6.6Ni0.18C0.7Mn0.5Si0.15N as-received

SUS304 (304) 18Cr9Ni0.022C1.03Mn0.38Si 1373 K©1.8 ks (WQ)

*R*

*a*
*P*

*P*
*A*

*P*

π*a*2_{= A}

**=**σ*av.* σ* _{zz}* σ σ

*xx*

σ*z*

σ*x*

σ*y*

σ*zz=*σ*+*σ*xx*

Equation (4) is assumed to have rotational symmetry about the longitudinal axis, and eq. (5) means · is equal to the equivalent stress by the von Mises plasticity condition. By using eqs. (4) and (5), the following equation is obtained for the axial stress.12,17)

·zz¼·þ·xx ð6Þ

This indicates that the stress at any point can be considered to be composed of a uniform true stress (·) and a nonuniform hydrostatic tension (·xx) as seen in Fig. 1. At the external

surface,·zzis equivalent to·because the stress component of

·xxis zero at the neck.12,13)Thus, the stress equation can be

described by the single component·xxthrough

d·xx

dx Rþ

a2_{}_{x}2
2a

þx_{a}·¼0 ð7Þ

wherexis the radial distance in the minimum section of the neck, and R is the radius of curvature of the neck proﬁle.

By solving eq. (7), ·xx and ·zz are given by the following

equations,

·xx¼·loga

2_{þ}_{2}_{aR}_{}_{x}2

2aR ð8Þ

·zz¼· 1þloga

2_{þ}_{2}_{aR}_{}_{x}2

2aR

ð9Þ

As seen in Fig. 1, the average stress (·av.) is described by,

·av:¼_{³}P_{a}_{2} ð10Þ

where Pis the load and is given by the following equation using the eq. (9),

P¼2³
Z_{a}

0

x·zzdx¼³·ða2þ2aRÞlog 1þ a 2R

ð11Þ

Therefore, a simple way to determine·is proposed by using the following equation,1214)

· ¼_{} ·av:

1þ2R

a

log

1þ a 2R

¼ P

³a2

1þ2R

a

log

1þ a 2R

ð12Þ

It is possible to use eqs. (3) and (12) to estimate ¾ and · up until the plastic deformation limit by using the values of

R, a, and P obtained in the stepwise tensile tests.17) _{The}

validation of eq. (12) was examined by Marshall et al.14)

who investigated the effect of the neck proﬁle radius on the

·¾curves.

4. Results and Discussion

4.1 Nominal stressstrain curves and their static tensile properties

Figure 2 shows representative nominal stressstrain curves of various metals and alloys. The mechanical properties obtained by static tensile tests are summarized in Table 2. Here, the yield strength for QT1, FP2 and ufg-FC steels means lower yield stress and those for other materials are 0.2% proof stress. In the present experimental results,

tempered martensite (QT3) displayed the largest strength, while aluminum (Al) showed the smallest strength. Figures 3 and 4 show uniform and total elongations as a function of normalized yield strength (·y/E) and tensile strength (·B/E)

whereEis Young’s modulus, respectively. The uniform and total elongations decreased with increasing·y/Eand·B/E.27)

In Figs. 2 and 3, the 304, 310S, 329J4L steels and Ti indicated a better balance of strength and elongation.

4.2 True stresstrue strain relationships up until the plastic deformation limit

Figure 5 shows estimated·¾relationships obtained by the
stepwise tensile tests and the Bridgman equation in various
metals and alloys.17)_{In this}_{ﬁgure, the curves are calculated}

by using eqs. (1) and (2), and plots of¾and·are estimated by eqs. (3) and (12). The·¾relationships with¾larger than approximately 1.0 were obtained, and the true strains at the plastic deformation limit (¾pdl) were more than three times the

uniform elongation in most cases. In the nominal stressstrain curves in Fig. 2, the uniform or total elongation typically became smaller with increasing strength. However, the estimated·¾ relationships with a wide · range or strength were obtained despite¾pdlbeing almost the same value (1.7).

In the comparisons between the tensile properties of ufg-FC and FP2, which were prepared from the same low carbon steel, the total elongation of ufg-FC was smaller than that of FP2 (Fig. 2), but ufg-FC showed a larger ¾pdl than FP2. On

the other hand, the strain rate increased after the necking started because the tensile test was conducted with the constant crosshead speed. Therefore, such the increase in strain rate seems to affect the·¾relationships after necking. Figure 6 shows the reduction in area as a function of (a) total elongation and (b) ¾pdl. The ¾pdl became larger as the

reduction in area increased, but this was not the case for the total elongation. This means that¾pdlis closely related to the

reduction in area (the minimum radius of the neck cross section) as indicated in eq. (3), but not to a change in the gage length. The ¾pdl is different from the total elongation

despite it being for the same material. Therefore, the reduction in area seems to play an important role in the ¾ after necking in round tensile specimens. The ¾pdl is also

0 200 400 600 800 1000 1200 1400

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

**Nominal str**

**ess, **

**s **

**/ MP**

**a**

**Nominal strain, *** e*
QT3

329J4L

QT1

304

310S

Al Cu Ti 329J1

FP2

430 ufg-FC

associated with the fracture stress.26) _{In the case of a}

reduction in area of more than 80%, the ¾pdl was

approximately 1.7, almost independent of the reduction in

area. It may be difﬁcult to estimate the · and ¾ when the reduction in area is greater than 80% in the present experimental procedures.

0 10 20 30 40 50 60 70

0 1 2 3 4 5 6 7

**Unif**

**o**

**rm elongation, **

ε**u**

** / %**

**Normalized yield strength, **σ_{y}**/E**** (x10-3 _{)}**

304

310S

329J4L 329J1 Al

Cu

QT3 FP1

430

Ti

FP2

QT2 QT1

ufg-FC (a)

0 1 2 3 4 5 6 7

**Normalized tensile strength, **σ_{B}**/E**** (x10-3 _{)}**

304

329J4L 329J1 310S

Cu

Al

Ti

430 FP1

QT3 FP2

QT2 QT1 ufg-FC (b)

Fig. 3 Uniform elongation as a function of normalized yield strength (a) and normalized tensile strength (b) in various metals and alloys. Table 2 Mechanical properties obtained by static tensile tests in various metals and alloys.

Material Yield strength (MPa)

Tensile strength (MPa)

Uniform elongation

(%)

Total elongation

(%)

Reduction in area (%)

Fracture stress (MPa)

Al 26 58 27.5 54.1 97.0 20

Cu 49 215 35.9 60.3 92.6 96

310S 192 505 45.0 66.5 88.5 259

FP1 251 470 20.0 34.0 60.7 358

FP2 341 522 18.5 34.0 69.4 309

QT1 701 763 6.9 20.0 68.8 398

QT2 1027 1073 3.8 14.0 58.6 677

QT3 1182 1307 3.0 14.0 56.1 756

ufg-FC 545 612 10.0 24.7 81.0 302

430 230 402 22.3 42.8 85.0 206

Ti 273 402 22.6 45.9 74.2 281

329J1 594 659 13.4 31.1 75.1 319

329J4L 730 834 18.2 34.8 75.3 399

304 190 599 65.2 77.2 81.9 324

0 10 20 30 40 50 60 70 80

0 1 2 3 4 5 6 7

**T**

**o**

**tal elongation, **

δ

** / %**

**Normalized yield strength, **σ_{y}**/E**** (x10-3 _{)}**

304

310S

329J4L

329J1 Al

Cu

QT3 FP1

430 Ti

FP2

QT2 QT1 ufg-FC (a)

0 1 2 3 4 5 6 7

**Normalized tensile strength, **σ_{B}**/E**** (x10-3 _{)}**

304

329J4L 329J1 310S

Cu

Al

Ti

430

FP1

QT3 FP2

QT2 QT1 ufg-FC (b)

Figure 7 shows loglog plots of the ·¾ relationships in Fig. 5. The ·¾ curves can be described by using the Hollomon equation,2,28)

·¼K¾n _{ð}_{13}_{Þ}

where K and n are constants determined from the intercept and slope of log(·)log(¾) plots. It can be seen that the slope of the log(·)log(¾) plots for materials other than bcc structures changes corresponding to the strain regimes. This means that a few constants are necessary for eq. (13) with strain regimes in order to describe the·¾relationships until just before fracture. The values ofKandn in the Hollomon equation for each region and the transition strains are summarized in Table 3. The transition strains were almost consistent with uniform elongation in most materials.

Figure 8 shows comparisons of the se, ·¾, and ·av.¾

relationships in the 310S, 329J4L, QT3, and ufg-FC steels. The ¾pdl for QT3 was approximately 0.8 whereas the total

elongation was about 0.1 (10%). As discussed in relation to
Fig. 6, it was found that ¾ is associated with the minimum
radius of the neck cross section. In the majority of
experimental results, necking started at an ¾ value
approx-imately double that of the uniform elongation.10)_{This means}
· is equal to·av.until¾ is double the uniform elongation in

most cases.

4.3 Tensile properties in terms of the true stress and true strain at the plastic deformation limit

Figure 9 shows the relationships between normalized true stress (·pdl/E) and true strain (¾pdl) at the plastic deformation

limit in various metals and alloys. The dashed lines in Fig. 9 are contour lines of the product of ·pdl/E and ¾pdl. In the

strengthelongation balance, as seen in Figs. 3 and 4, the
uniform and total elongations decrease with increasing yield
and tensile strength, respectively.26,27) _{However, the good}

balance in Fig. 9 can be achieved in the case that both·pdl/E

and ¾pdl are large. Therefore, the 304 and 329J4L steels

indicated a superior·pdl/E¾pdlbalance. Figure 10 shows the

normalized yield strength (·y/E) against ¾pdl. Not only the ·pdl but also the ·y is important in the ·¾ relationships up

until the plastic deformation limit as the stress at which the plastic deformation starts. By replacing·pdl/Ewith·y/E, the

strength¾pdlbalance differs between Figs. 9 and 10. The 304

and 310S steels, which showed the good balance in Fig. 9, indicated almost the same ·y/E¾pdl balance as the 430 and

FP1 steels. On the other hand, the 329J4L, 329J1, QT, and ufg-FC steels indicated a better balance in Fig. 10. It is interesting that the·y/E¾pdlbalance of the QT steels is good

despite the total elongation being small. The duplex stainless steels of 329J4L and 329J1 indicated a better balance in both Fig. 9 and Fig. 10. This seems to be associated with

0 20 40 60 80 100

0 10 20 30 40 50 60 70 80

**Reduction in ar**

**ea, **

φ

** / %**

**Total elongation, **δ** / %**
QT3

QT2 QT1 FP2

FP1

304 329J1

329J4L 430

Ti 310S Al

Cu ufg-FC

(a)

0 0.5 1 1.5 2

ε** at the plastic deformation limit, **ε
* pdl*
QT3

FP2

Ti 304

329J4L 329J1

430 Al

310S Cu

QT2 QT1

FP1 ufg-FC (b)

Fig. 6 Reduction in area as a function of total elongation (a) and true strain at the plastic deformation limit (b).

100 1000

0.01 0.1 1

**T**

**rue str**

**ess, **

σ

** / MP**

**a**

**True strain, ε**
Al
Cu

430 ufg-FC 329J4L

QT3

Ti

FP2

304

Fig. 7 Log true stress vs. log true strain plots in various metals and alloys. 0

400 800 1200 1600

0 0.5 1 1.5 2

**T**

**rue str**

**ess, **

σ

** / MP**

**a**

**True strain, **ε

Al Cu

430 310S 304

ufg-FC 329J4L

QT3

Ti

FP2

dual-phase strengthening8) _{and grain reﬁnement }

strengthen-ing3,4) _{because both steels are dual-phase steels with a grain}

size of less than 10 µm. The grain size is also associated with
the ·¾ relationship of the ufg-FC steel. It should be noted
that not only · and the strength but also ¾ increased with
decreasing grain size. Good ductility2326) _{and bending}

properties9) of ultraﬁne-grained steels have been reported, and these experimental results are difﬁcult to explain in terms of the total elongation but they are able to be discussed using the ·¾ relationship up until the plastic deformation limit. The estimated ·¾ relationships in the ufg-FC steel seem to indicate that grain reﬁnement strengthening can improve ·

and ¾ up until the plastic deformation limit. Therefore, the two strengthening mechanisms, dual-phase strengthening and grain reﬁnement strengthening, are effective for the improve-ment of·pdland¾pdl.26)As discussed above, the summary for

the tensile properties in terms of·pdland¾pdlwas found to be

Table 3 Values forKandnin Hollomon equation and transition strains in various metals and alloys.

Material Region 1 Region 2 Region 3

K1 n1 ¾12* K2 n2 ¾23* K3 n3

Al 106.3 0.25 0.28 118.9 0.353 ® ® ®

Cu 530.9 0.515 0.33 402.7 0.267 ® ® ®

310S 721.3 0.30 0.08 1209.5 0.512 0.70 1065.7 0.303

FP1 885.0 0.20 ® ® ® ® ® ®

FP2 865.0 0.19 ® ® ® ® ® ®

QT1 1040 0.090 ® ® ® ® ® ®

QT2 1295 0.045 ® ® ® ® ® ®

QT3 1550 0.040 ® ® ® ® ® ®

ufg-FC 977 0.176 ® ® ® ® ® ®

430 1102 0.21 ® ® ® ® ® ®

Ti 759 0.283 ® ® ® ® ® ®

329J1 926.8 0.107 0.17 1148.2 0.228 ® ® ®

329J4L 1151.3 0.105 0.11 1448.8 0.209 ® ® ®

304 845.7 0.322 0.13 1474.0 0.598 0.56 1316.1 0.397

*¾12and¾23are the transition strains between regions 1 and 2 and between regions 2 and 3, respectively.

0 500 1000 1500 2000

0 0.5 1 1.5 2

**Str**

**ess, **

**s**

** or **

σ

** / MP**

**a**

**Strain, ****e**** or **ε

QT3 329J4L

310S

ufg-FC

σ*av*.–ε
σ–ε

*s*–*e*

Fig. 8 Comparisons of nominal stress (s), true stress (·) and average stress (·av.) as a function of nominal (e) or true strain (¾) in the SUS310S,

SUS329J4L, QT3 and ufg-FC steels.

0 2 4 6 8 10

0 0.5 1 1.5 2

**Normalized **σ**at the plastic deformation limit,**σ_{pdl}**/E**** (**x**10-3)**

ε

** at the plastic def**

**ormation limit, **

ε**pdl**_{Al}

Cu 430

Ti

QT2 QT3 QT1

FP2 FP1

310S

304

329J4L 329J1

2.5 5 7.5 10 15

ufg-FC 12.5

Fig. 9 Normalized true stresstrue strain relationships at the plastic deformation limit in various metals and alloys.

0 0.5 1 1.5 2

0 1 2 3 4 5 6 7

ε

** at the plastic def**

**ormation limit, **

ε**pdl**

**Normalized yield strength, **σ_{y}**/E**** (**x**10-3)**
304

329J4L 329J1 310S

Al

Cu 430

FP1

QT3 Ti

1 2 3 4 5 FP2

QT2 QT1

ufg-FC

very different from the conventional tensile properties using the tensile strength, uniform elongation, and total elongation. The ·pdl/E¾pdl balances, as seen in Fig. 9, seem to be

associated with work-hardening behavior in their estimated

·¾ relationships. The ·pdl/E¾pdl balance can be discussed

in terms of the hardening behavior. Here, the work-hardening rate at the plastic deformation limit ((d·/d¾)pdl)

was calculated by using the constants in Table 3 as

d· d¾ pdl

¼Kn¾pdln1 _{ð}_{14}_{Þ}

Figure 11 shows the products of · and ¾ at the plastic deformation limit (·pdl©¾pdl) as a function of (d·/d¾)pdl in

various metals and alloys. As can be seen, the ·pdl©¾pdl

values are dependent on (d·/d¾)pdl and became larger with

increasing (d·/d¾)pdl. At this time, the ·pdl and ¾pdl values

for the various metals and alloys cannot be summarized by (d·/d¾)pdl. Therefore, the ·pdl©¾pdl values are closely

associated with the work-hardening behavior. It is concluded that an increasing work-hardening rate plays an important role in the accomplishment of a superior·pdl©¾pdlvalue.

5. Conclusions

In this study, the true stress (·) and true strain (¾) up until the plastic deformation limit (just before fracture) were estimated by the stepwise tensile tests and Bridgman equation in various metals and alloys with different crystal structures. The following conclusions were obtained from this study.

(1) The ·¾relationships up until the plastic deformation limit with¾greater than 1.0 were obtained for various metals and alloys. A few values ofKandnare necessary to describe the·¾curves using the Hollomon equation for materials that have a structure other than bcc.

(2) In the relationship between the true stress (·pdl) and

true strain (¾pdl) at the plastic deformation limit, SUS304 and

SUS329J4L indicated a better normalized ·pdl¾pdl balance.

On the other hand, SUS329J4L, SUS329J1, QT, and ufg-FC steels showed superior results in the normalized yield strength¾pdlbalance.

(3) The estimated·¾relationship for the ultraﬁne-grained low-carbon steel (ufg-FC) suggests that grain reﬁnement strengthening can improve · and ¾ up until the plastic deformation limit.

(4) The value of ¾pdl became larger with the reduction in

area and a decrease in the fracture stress. The products of·pdl

and¾pdlbecame larger with increasing work-hardening rate at

the plastic deformation limit.

Acknowledgments

The authors are grateful to Professor J. Yanagimoto of The University of Tokyo and Dr. A. Yanagida of Tokyo Denki University for their valuable discussions. This study was ﬁnancially supported by the Grant-in-Aid for Scientiﬁc Research on Innovative Area,“Bulk Nanostructured Metals”, through MEXT, Japan (contract No. 22102005), and the support is gratefully appreciated.

REFERENCES

1) C. W. MacGregor: J. Franklin Inst.238(1944) 111135. 2) J. H. Hollomon: Trans. AIME162(1945) 268290.

3) N. Tsuji, R. Ueji, Y. Minamino and Y. Saito: Scr. Mater.46(2002) 305 310.

4) R. Song, D. Ponge, D. Raabe, J. G. Speer and D. K. Matlock: Mater. Sci. Eng. A441(2006) 117.

5) Y. Fukuda, K. Oh-ishi, Z. Horita and T. G. Langdon: Acta Mater.50

(2002) 13591368.

6) Z. Horita: J. Jpn. Inst. Light Metals60(2010) 134141.

7) A. Ohmori, S. Torizuka and K. Nagai: ISIJ Int.44(2004) 10631071. 8) N. Tsuchida, H. Masuda, Y. Harada, K. Fukaura, Y. Tomota and K.

Nagai: Mater. Sci. Eng. A488(2008) 446452.

9) K. Miyata, M. Wakita, S. Fukushima and T. Tomida: Proc. the 2nd Int. Symp. on Steel Science (ISSS-2009), (2009) pp. 115120.

10) G. E. Dieter: Mechanical Metallurgy, (McGraw-Hill, London, 1988) pp. 291295.

11) ASM Handbook: Mechanical Testing and Evaluation vol.8, (ASM International, Ohio, 2000) pp. 105106.

12) P. W. Bridgman: Studies in Large Plastic Flow and Fracture, (McGraw-Hill, New York, 1952) pp. 937.

13) P. W. Bridgman: Trans. ASM32(1944) 553574.

14) E. R. Marshall and M. C. Shaw: Trans. ASM44(1952) 705725. 15) W. F. Hosford: Mechanical Behavior of Materials, (Cambridge

University Press, New York, 2005) pp. 4447. 16) J. Aronofsky: J. Appl. Mech.18(1951) 7584.

17) K. Enami and K. Nagai: Tetsu-to-Hagane91(2005) 712718. 18) K. Enami and K. Nagai: Tetsu-to-Hagane91(2005) 285291. 19) K. Enami, K. Nagai, S. Torizuka and T. Inoue: Tetsu-to-Hagane 91

(2005) 769774.

20) G. Lankford and M. Cohen: Trans. Am. Soc. Met.62(1969) 623638. 21) T. Csanadi, N. Q. Chinh, J. Gubicza and T. G. Langdon: Acta Mater.59

(2011) 23852391.

22) T. Inoue, S. Torizuka and K. Nagai: Mater. Sci. Technol.17(2001) 13291338.

23) Y. Kimura, T. Inoue, F. Yin and K. Tsuzaki: Science320(2008) 1057 1060.

24) T. Inoue, F. Yin, Y. Kimura, K. Tsuzaki and S. Ochiai: Metall. Mater. Trans.41A(2010) 341355.

25) S. Torizuka, E. Muramatsu, S. V. S. N. Murty and K. Nagai: Scr. Mater.

55(2006) 751754.

26) N. Tsuchida, S. Torizuka, K. Nagai and R. Ueji: Tetsu-to-Hagane96

(2010) 4250.

27) E. De Moor, P. J. Gibbs, J. G. Speer, D. K. Matlock and J. G. Schroth: AIST Trans.7(2010) 133144.

28) J. H. Hollomon: Metals Technol.12(1945) 122.

0 500 1000 1500 2000 2500 3000

0 50 100 150 200 250 300 350 400

σ **pdl**

**x**

ε **pdl**

** / MP**

**a**

**Work-hardening rate at **σ

* pdl*, (

*σ*

**d***ε*

**/d****)**

**pdl****/ MPa**304

329J4L

329J1 310S

Al Cu

Ti FP2 QT1

430

FP1 QT2

QT3

ufg-FC

Fig. 11 The products of true stress and true strain (·pdl©¾pdl) vs.