## Prediction on Nominal Stress-Strain Curve of Isotropic Polycrystal

## Ti-15-3-3-3 Sheet by FE Analysis

### Long Li, Fuxing Yin and Kotobu Nagai

National Institute for Materials Science, Tsukuba 305-0047, Japan

In order to predict the nominal stress-strain curve of an isotropic polycrystal Ti-15V-3Al-3Cr-3Sn (hereafter, Ti-15-3(ST)) sheet in tension test, three-dimensional ﬁnite element (FE) analysis is elucidated considering three boundary conditions. According to the constraint at both ends of specimen, three boundary conditions are a simulated case based on the empirical data, a full case and a free case. In the simulated case, the nominal stress-strain curve can be well predicted until the fracture strain. Maximum load point does not mean the termination of uniform deformation in the present study. The onset of localized necking is intensively discussed and the origin of localized necking is concluded to be

not due to the plastic instability but due to the deformation constraint at both ends of specimen. [doi:10.2320/matertrans.M2010033]

(Received January 26, 2010; Accepted May 12, 2010; Published September 8, 2010)

Keywords: titanium-15 vanadium-3 aluminum-3 chromium-3 tin sheet, tensile test, stress-strain curve, ﬁnite element method, uniform deformation

1. Introduction

The uniform deformation until the onset of localized necking proceeds under the uniaxial stress state. However, after the onset of localized necking, the stress state changes from the uniaxial one to biaxial one1,2) and further compli-cated one in the tensile test.3,4)

Under a load-controlled mode of deformation, the metal seems to proceed into fracture directly at maximum load (Pmax) as shown in Fig. 1(a). However, under a

displacement-controlled mode of deformation, on the other hand, the metal
shows a diﬀerent load-displacement curve as shown in
Fig. 1(b).1–4) _{In the ﬁrst stage, the load increases with an}

increase in displacement from yielding to Pmax. AfterPmax,

the load decreases gradually with an increase in
displace-ment.2)_{If the fracture does not intervene, localized necking}

may then occur generally at the center of the gage length. The
localized necking leads to the localized through-thickness
thinning in the plane of a sheet specimen, and then the ﬁnal
fracture at the localized necking point.1,2) _{Since the onset}

of localized necking cannot be theoretically predicted, the authors may set the second stage fromPmax to the onset of

localized necking and the third stage from the onset of localized necking to ﬁnal fracture. Plastic deformation limit (PDL) as the onset of ductile fracture5)is also proposed in the third stage as in Fig. 1.

The shape of a load-displacement curve or a nominal
stress-strain curve changes according to deformation mode
such as tension, compression and bend even for the same
ductile metal with an identical true stress-strain relation. In
other words, the boundary condition deﬁnitely aﬀects the
shape of the nominal stress-strain curve. Recently, ﬁnite
element (FE) analysis has been successful in prediction of the
deformation behavior in tension test,1–4)_{compressive test,}6,7)

and torsion test.8) Based on the FE idea, it can be easily admitted that the shape of the nominal stress-strain curve can be perfectly predicted if the true stress-strain relation is given. However, quite interestingly, few have addressed this idea considering various boundary conditions, which hap-pens to be the principal purpose of the present paper. When

no grip constraint is supposed in thickness and width directions for a sheet specimen, the uniform deformation can be kept until the cross section becomes nothing. In real cases, however, the grip constraint is inevitable and may become one of the origins to introduce the non-uniform

Maximum Load

Fracture

Load

Displacement (a) Load-controlled mode

Yielding

Yielding

Maximum Load

Localized necking

Fracture

1st Stage

2nd Stage

3rd Stage

Load

Displacement PDL (b) Displacement-controlled mode

Fig. 1 Schematic plot of the load-displacement curve of a ductile metal

sheet in tension test (a) load-controlled mode; (b) displacement-controlled mode.

deformation. For the simpliﬁcation, the volume constancy is admitted or the defect formation like the void evolution is neglected.

An isotropic polycrystal Ti-15-3(ST) alloy is one of good examples that show the three stage deformation.9–13) In the present paper, the nominal stress-strain curve of Ti-15-3(ST) alloy is predicted in the whole tension test by means of FE analysis with various boundary conditions and the prediction result is compared with the empirical data.

2. Boundary Conditions

Three boundary conditions have been discussed as schematically shown in Fig. 2 and summarized in Table 1 and Fig. 3. As shown in Fig. 3, the simulated case is calculated according to empirical data.14) The free case assumes noy- and z-direction constraint at both ends of the parallel part of specimen, neglecting the occurrence of any inhomogeneous deformation. The full case assumes full grip constraint alongy- andz-direction at both ends of the parallel part of specimen. According to Table 1, the onset of

localized necking (Plocal) can be denoted as the deviation

point of the simulated case from the free case on the nominal stress-strain curve.

3. Fundamental Deﬁnition

Strain, measured most commonly with extensometers, can be expressed as:

"n¼ ll0

l0

ð1Þ

where "n is nominal strain, l0 refers to initial gage length,

lgage length of the deformed specimen. True strain"is deﬁned as:

"¼lnA0

A ð2Þ

whereA0 is the initial cross-sectional area of specimen,Ais

the cross-sectional area of deformed specimen.

A nominal stressncan be expressed as force per unit area

of cross section:

(a) Free case (b) Full case (c) Simulated case

*L0*

*P * *P *

Y X

Z *L0*

*P* *P*

*L0*

*P* *P*

Constraint based on data Full

Constraint

Fig. 2 Three boundary conditions in FE analysis (a) free case; (b) full case;

(c) simulated case based on empirical data (L0¼35mm).

### Case

### x-direction (Length)

*L0*

### =35mm

### y-direction (Width)

*w0*

### /2=3mm

### z-direction (Thickness)

*t0*

### =2mm

### Free

**0** **50** **100 150 200 250**
**0.0**
**0.7**
**1.4**
**2.1**
**2.8**
**3.5**
**4.2**
**4.9**
0
2
4
6
8
10
12
14

Percent displacement (X),%

Displacement (X),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.06**
**0.12**
**0.18**
**0.24**
**0.30**
**0.36**
**0.42**
0
2
4
6
8
10
12
14

Percent displacement (Y),%

Displacement (Y),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.04**
**0.08**
**0.12**
**0.16**
**0.20**
**0.24**
**0.28**
0
2
4
6
8
10
12
14

Percent displacement (Z),%

Displacement (Z),mm

Time,s

### Full

**0** **50** **100 150 200 250**
**0.0**
**0.7**
**1.4**
**2.1**
**2.8**
**3.5**
**4.2**
**4.9**
0
2
4
6
8
10
12
14

Percent displacement (X),%

Displacement (X),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.06**
**0.12**
**0.18**
**0.24**
**0.30**
**0.36**
**0.42**
0
2
4
6
8
10
12
14

Percent displacement (Y),%

Displacement (Y),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.04**
**0.08**
**0.12**
**0.16**
**0.20**
**0.24**
**0.28**
0
2
4
6
8
10
12
14

Percent displacement (Z),%

Displacement (Z),mm

Time,s

### Simulated

**0** **50** **100 150 200 250**
**0.0**
**0.7**
**1.4**
**2.1**
**2.8**
**3.5**
**4.2**
**4.9**
0
2
4
6
8
10
12
14

Percent displacement (X),%

Displacement (X),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.06**
**0.12**
**0.18**
**0.24**
**0.30**
**0.36**
**0.42**
0
2
4
6
8
10
12
14

Percent displacement (Y),%

Displacement (Y),mm

Time,s **0** **50** **100 150 200 250**

**0.00**
**0.04**
**0.08**
**0.12**
**0.16**
**0.20**
**0.24**
**0.28**
0
2
4
6
8
10
12
14

Percent displacement (Z),%

Displacement (Z),mm

Time,s

Fig. 3 Boundary conditions in details on the nodes at both ends of specimen in FE analysis (L0¼35mm,t0¼2mm,w0¼6mm).

Table 1 Three boundary conditions in FE and experimental analysis.

Case

Grip constraint

iny- and

z-direction Uniform Deformation Maximum load point Localized necking Void formation

Free No Full Yes No No

Full Yes Partly Yes No No

Simulated Yes Partly Yes No No

n¼ P A0

ð3Þ

where P is the tensile load. If the reduction of the cross-sectional area is large, the nominal stress deﬁnition becomes inappropriate. An average stressdeﬁnition should use the instantaneous cross-sectional area:

¼P

A ð4Þ

The deﬁnition of the average stress is based on the instantaneous material conﬁguration. If the volume of the specimen is conserved during tension test, the true stress-strain relation can be obtained by the nominal stress-stress-strain curve as follows:

¼nð1þ"nÞ; "¼lnð1þ"nÞ ð5Þ

It is important to note that eq. (5) holds only for uniform deformation. Once deformation ceases to be uniform, only average stress can be measured and the stress distribution cannot be determined empirically.15)

4. Experimental

4.1 Uniaxial tension test

A 15-3(ST) alloy with a nominal composition of Ti-15V-3Cr-3Sn-3Al was prepared as 2.0 mm thick sheets. A solution treatment for 900 s at 1123 K in a vacuum furnace was carried out to obtain a homogeneous microstructure. Tensile test was performed using a universal testing machine with an extensometer (a gage length l0 of 17.5 mm) and a

gage width of 6.0 mm. The test proceeds at a constant
crosshead speed of 1 mm/min (the initial strain rate of
9:5104_{s}1_{) at room temperature. The specimen }

geom-etry is given in Fig. 4. Interrupted tensile tests were also carried out to measure the cross-sectional size of deformed specimen; the tensile test was interrupted at 3.1%, 7.2%, 9.2%, 12.4% and 18.7% (corresponding to fracture) in elongation based on the indication of extensometer, corre-sponding to the nominal plastic strain of 0.03, 0.07, 0.09, 0.12, and 0.19 (fracture strain), respectively, and hereafter these strains are named as interrupted strains.

True stress-strain relation employed in the FE prediction is approximated from the empirical data in the strain range between 0.03 and 0.08 as follows:14)

¼848:2"0:03 ðR2 ¼0:958Þ ð6Þ

The pole ﬁgure and inverse pole ﬁgure reveal that there is no obvious texture in the solution-treated condition, and the isotropic deformation of the alloy can therefore be guaran-teed in the FE prediction.14)

4.2 FE analysis

In the present paper, an isotropy plasticity model and the
Mises function implemented in the commercial MSC.MARC
2005 software were used. A half discrete model was adopted
and the three-dimensional ﬁnite element mesh is shown in
Fig. 5.14) _{The spatial non-uniform ﬁnite element mesh is}

chosen and the mesh system has 52, 4 and 4 divisions in the

x-,y- andz-directions, respectively. Each element is chosen as 7-type element with 8-node particularly suited to large strain analysis.

As shown in Fig. 2 and Fig. 3, in the case of the free case, the nodes on bothx¼0mm section andx¼35mm section are free iny- andz-direction; for the full case, the nodes on thex¼0mm section have no displacement in all directions but the nodes onx¼35mm section have been ﬁxed alongy -and z-direction and the displacement along x-direction is controlled by moving the nodes based on the empirical data.14)The material parameters obtained by tensile test and the simulation details used in FE analysis are listed in Table 2.

5. Results and Discussion

5.1 Prediction of nominal stress-strain curve

The predicted nominal stress-strain curves and the em-pirical one are shown in Fig. 6(a). The predicted ones are drawn until the strain of about 0.19, which corresponds to the fracture strain (nominal) for the empirical one.

Except for the free case, the predicted curves are well matched with the empirical one in over the strain range investigated. The diﬀerence in the predicted stress is very small between the full case and the simulated case. In this respect, whether the full case or the simulated case has no signiﬁcant diﬀerence in the prediction of the nominal stress-strain curve.

Fig. 4 Tensile specimen geometry with a rectangular cross-section (unit:

mm).

*x*

*y*

*z*

*L*0=35mm

*w*0/2

*t0*

Fig. 5 Mesh system and boundary conditions in three-dimensional FE

analysis (t0¼2mm,w0¼6mm).

Table 2 Material parameters and simulated details used in FE analysis.

Material

Yield

Strength Poisson

ratio

Young’s

Modulus Characteristic

parameter (MPa) (GPa)

720.0 0.33 76.5 Isotropic

Simulation details

Element number

Node number

Step number

Step length

Figure 6(b) demonstrates the ratio of predicted nominal stress to the empirical one as a function of nominal strain. The error is small, less than 10% at the maximum, in the strain range investigated. Especially in the nominal strain range between 0.03 and 0.12, the error is almost 0%. Interestingly, in the similar range, the free case also shows the best conformity with other cases. Because the true strain-stress relation used in the simulation is derived from the part in the strain range between 0.03 and 0.08, the good prediction in this strain range is warranted. Therefore, the interesting point is that the good prediction can be obtained further over that strain range.

In the beginning stage of deformation, however, the error becomes large. In the present alloy, the spiky load peak appears at the nominal strain of 0.01 on the empirical curve. And the empirical load becomes highest here. The question is whether this peak corresponds to Pmax in Fig. 1. N.

Stefansson et al. attribute this peak to the originally low
density of mobile dislocation.10)_{Further, it is reported the }

Ti-15-3(ST) alloy shows the uniform deformation even after this
peak under given strain rates.9–11)_{Actually the eq. (6) leads to}

the identical maximum load point at the nominal strain of
0.03 for all the boundary conditions as shown in Fig. 6(a).
Accordingly, the maximum load point is not aﬀected by
boundary conditions in the present study where the gage
length is ﬁxed. W. J. Danet al.16)_{studied the eﬀect of gage}

length on the strain corresponding to maximum load point and the results showed thatPmaxalmost remain unchanged at

diﬀerent gage lengths. Hence, in the present paper, it is concluded that the spiky peak diﬀerent from the maximum load point deteriorates the prediction accuracy, and ﬁnally

Pmax appears at the nominal strain of 0.03 as shown in

Fig. 6(a).

5.2 Onset of localized necking

The free case assumes the fully uniform deformation. The nominal stress for all the cases is almost identical until the strain of 0.12. At the strain more than 0.12, the nominal stress for the free case becomes higher than that for other cases. Table 3 lists the minimum cross-section size meas-ured, the average stress and the true strain calculated by eqs. (4) and (2) at given interrupted strain, respectively. The average stress-true strain relations in the present alloy are shown in Fig. 7. The empirical average stresses at the interrupted strain of 0.03, 0.07, 0.09 and 0.12 are almost identical to the average stresses for the full case and the simulated case with a relative error less than 2%. No diﬀerence in the average stress between the full case and the

**0.00** **0.05** **0.10** **0.15** **0.20**

**0.9**
**1.0**
**1.1**
**1.2**

**Free vs Empirical**
** Full vs Empirical**
** Simulated vs Empirical**

**Predicted stress/Empirical stress**

**Nominal strain**

(b) (a)

**0.00** **0.05** **0.10** **0.15** **0.20**

**0**
**100**
**200**
**300**
**400**
**500**
**600**
**700**

**800** _{P}

local

**Free**
**Full**
**Simulated**

**Empirical**

**Nominal stress, MPa**

**Nominal strain**

**P**_{max}

Fig. 6 The predicted nominal stress-strain curves and the empirical one (a) and the ratio of predicted nominal stress to the empirical one as

a function of nominal strain (b).

Table 3 Details at minimum cross-section with diﬀerent interrupted strains.

Position Interrupted

Strain

Measured width (mm)

Measured thickness (mm)

Calculated area

(mm2_{)}

Load (N)

True strain by eq. (2)

Average stress (MPa) by eq. (4)

A 0.03 5.95 1.97 11.7 9000 0.02 768

B 0.07 5.78 1.92 11.1 8738 0.08 787

C 0.09 5.60 1.86 10.4 8563 0.14 822

D 0.12 5.41 1.80 9.7 8413 0.21 864

**0.0** **0.2** **0.4**

**0**
**100**
**200**
**300**
**400**
**500**
**600**
**700**
**800**
**900**
**1000**

**Full / Simulated**
**Empirical**

**B** **C**

**D**

**Average stress ,MPa**

**True strain**
**A**

**0.1** **0.3**

Fig. 7 Average stress-true strain curve of isotropic polycrystal Ti-15-3

simulated case is detected. This postulates again the validity of the present prediction at least in the nominal strain range up to 0.12.

H. Moriya et al.9) _{compared the stress-strain relations}

based on the eq. (4) and the eq. (5) and pointed out that the deviation in stress between two relations occurred at the strain between 0.10 and 0.15 for the Ti-15-3(ST) alloy. And they correlated the deviation with the occurrence of localized necking. Thus, the strain of 0.12 can be denoted as the onset of localized necking orPlocal when the overall shape of the

nominal stress-strain curve is discussed. This means the uniform deformation occurs not only in the 1st stage but also in the 2nd stage in Fig. 1(b).

5.3 Plastic deformation limit

The microscopic observations on steels show that ductile
cracking occurs associated with the growth and coalescence
of a large number of micro-voids generated in the localized
necking region and a rapid load decrease occurs just before
the ﬁnal ductile fracture.17)_{K. Enami}_{et al.}_{deﬁned the onset}

point of ductile fracture as the plastic deformation limit
(PDL).5)_{In other words, the PDL means the onset of a very}

rapid decrease in load or stress.5,17)

In the present paper, a rapid decrease in the nominal stress is not obviously observed by the strain of approximately 0.18. Hence, it is hard to determine the PDL. After thePlocal,

the precision of prediction gradually deteriorates with an increase in strain. It is believed that the precision of nominal stress prediction should be more improved when the fracture criterion can be applied in FE analysis.

5.4 Prediction of equivalent strain distribution

Predicted strain contours at interrupted strains are plotted for the simulated case in Fig. 8. It clearly demonstrates the development of non-uniformity of deformation; the max-imum strains are always found at the center of tensile specimen and the minimums are at both ends. A uniform strain part appears around the center of specimen and becomes narrower with an increase in interrupted strain, shown in Fig. 8.

Constraint at both ends retards the deformation at both ends; the strain gradient appears even at a small strain of 0.03 (Fig. 8(a)). The strain gradient gradually increases in terms of both magnitude and range and the uniform strain part shrinks with an increase in interrupted strain. At the interrupted strain of 0.03, the uniform strain part prevails as

shown in Fig. 8(a). However, at the strain of 0.12 (Fig. 8(d)), the uniform strain part remains scarcely at the center of specimen. Eventually, the center of tensile specimen turns into a localized strain part when the interrupted strain reaches 0.19 (Fig. 8(e)). An abrupt change is considered to occur at around the strain of 0.12. The present method predicts the change of specimen shape and applied load without any premise of localized deformation. Namely, the prediction spontaneously leads the strain contours and the shape in the specimen as shown in Fig. 8. The present result implies that the localized deformation can happen only due to the specimen constraint or the boundary condition.

In the present paper, the central part of 17.5 mm in length is a measure to deﬁne the uniform deformation. In this sense, it can be said that the uniform deformation keeps until at least the interrupted strain of 0.09. This well comes along with the previous studies for the Ti-15-3(ST) alloy. If the existence of some uniform part at the center of the gage length is taken as a measure for the uniform deformation, the present FE prediction can tell that the uniform deformation may terminate around at the strain of 0.12. This suggests that the Plocal depends on the empirical parameters such as

specimen dimension especially gage length.

6. Conclusions

(1) Nominal stress-strain curve until fracture can be well predicted by FE analysis for a isotropic polycrystal Ti-15-3(ST) sheet, which is veriﬁed by comparison with the empirical data. Full case and simulated case based on the empirical data show almost no diﬀerence on the prediction of nominal stress-strain curve.

(2) Constraint cases including full and simulated case show an obvious deviation of nominal stress from free case at the strain of approximately 0.12. When the existence of some uniform part at the center of the gage length is taken as a measure for the uniform deformation, the present FE prediction can tell that the uniform deformation may terminate around at the strain of 0.12. Thus, the strain of 0.12 can be denoted as the onset of localized necking.

(3) Maximum load point does not mean the termination of uniform deformation in the present study. The origin of localized necking is concluded to be not due to the plastic instability but due to the deformation constraint at both ends of specimen when microstructural inhomogeneity and temperature rise due to deformation heating are neglected.

0.46 0.01

0.03

0.01 0.03

### (a)

0.01 0.06 0.06

### (b)

0.08

0.01 0.08

### (c)

0.13 0.01

### (d)

### (e)

0.13

Y

X Z

Fig. 8 Eﬀective strain contours predicted by FE analysis with simulated case at the diﬀerent interrupted strains: (a) 0.03; (b) 0.07;

Acknowledgment

This study is conducted as part of the LISM (Layer-Integrated Steels and Metals) Project funded by Ministry of Education, Culture, Sports, Science and Technology of Japan. The authors gratefully acknowledge Dr. Tadanobu Inoue for his helpful discussion.

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