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

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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

3

1Graduate School of Engineering, University of Hyogo, Himeji 671-2280, Japan 2National Institute for Materials Science, Tsukuba 305-0047, Japan

3TOPY 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 stress­nominal

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) Bridgman12,13)

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)

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 ofR, 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, ·xxyy, ·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

2þ 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

πa2= A

=σav. σzz σ σxx

σz

σx

σy

σzz=σ+σxx

(3)

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

a2x2 2a

þxa·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

2þ2aRx2

2aR ð8Þ

·zz¼· 1þloga

2þ2aRx2

2aR

ð9Þ

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

·av:¼³Pa2 ð10Þ

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

P¼2³ Za

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 thisfigure, 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

(4)

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,

εu

/ %

Normalized yield strength, σy/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, σB/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, σy/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, σB/E (x10-3)

304

329J4L 329J1 310S

Cu

Al

Ti

430

FP1

QT3 FP2

QT2 QT1 ufg-FC (b)

(5)

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 ð13Þ

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

(6)

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

σav.–ε σ–ε

se

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 σat the plastic deformation limit,σpdl/E (x10-3)

ε

at the plastic def

ormation limit,

εpdl 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,

εpdl

Normalized yield strength, σy/E (x10-3) 304

329J4L 329J1 310S

Al

Cu 430

FP1

QT3 Ti

1 2 3 4 5 FP2

QT2 QT1

ufg-FC

(7)

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 ð14Þ

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.

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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, (dσ/dε)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.

Figure

Table 1Summary of materials in the present study.

Table 1.

Summary of materials in the present study . View in document p.2
Fig. 1Schematic illustration for stress distribution in the neck of a roundtensile test specimen.

Fig 1.

Schematic illustration for stress distribution in the neck of a roundtensile test specimen . View in document p.2
Fig. 2Nominal stress­strain curves obtained by static tensile tests invarious metals and alloys.

Fig 2.

Nominal stress strain curves obtained by static tensile tests invarious metals and alloys . View in document p.3
Figure 5 shows estimated ·­¾ relationships obtained by the
Figure 5 shows estimated relationships obtained by the. View in document p.3
Table 2Mechanical properties obtained by static tensile tests in various metals and alloys.

Table 2.

Mechanical properties obtained by static tensile tests in various metals and alloys . View in document p.4
Fig. 4Total elongation as a function of normalized yield strength (a) and normalized tensile strength (b) in various metals and alloys.

Fig 4.

Total elongation as a function of normalized yield strength a and normalized tensile strength b in various metals and alloys . View in document p.4
Fig. 5True stress­true strain relationships obtained by the stepwise tensiletests in various metals and alloys.

Fig 5.

True stress true strain relationships obtained by the stepwise tensiletests in various metals and alloys . View in document p.5
Fig. 7Log true stress vs. log true strain plots in various metals and alloys.

Fig 7.

Log true stress vs log true strain plots in various metals and alloys . View in document p.5
Figure 9 shows the relationships between normalized true­¾pdl balance as the 430 and ·y/E­¾pdl balance of the QT steels is good ·pdldespite the total elongation being small
Figure 9 shows the relationships between normalized true pdl balance as the 430 and y E pdl balance of the QT steels is good pdldespite the total elongation being small. View in document p.5
Fig. 10True strain at the plastic deformation limit as a function ofnormalized yield strength in various metals and alloys.

Fig 10.

True strain at the plastic deformation limit as a function ofnormalized yield strength in various metals and alloys . View in document p.6
Table 3Values for K and n in Hollomon equation and transition strains in various metals and alloys.

Table 3.

Values for K and n in Hollomon equation and transition strains in various metals and alloys . View in document p.6
Fig. 9Normalized true stress­true strain relationships at the plasticdeformation limit in various metals and alloys.

Fig 9.

Normalized true stress true strain relationships at the plasticdeformation limit in various metals and alloys . View in document p.6
Fig. 8Comparisons 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.

Fig 8.

Comparisons of nominal stress s true stress and average stress av as a function of nominal e or true strain in the SUS310S SUS329J4L QT3 and ufg FC steels . View in document p.6
Fig. 11The products of true stress and true strain (·pdl © ¾pdl) vs. work-hardening rate at the plastic deformation limit in various metals and alloys.

Fig 11.

The products of true stress and true strain pdl pdl vs work hardening rate at the plastic deformation limit in various metals and alloys . View in document p.7