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Prediction of Nominal Stress-Strain Curves

of a Multi-Layered Composite Material by FE Analysis

Long Li, Satoshi Iwasaki, Fuxing Yin and Kotobu Nagai

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

In order to predict the nominal stress-strain (S-S) curve in tensile test for a multi-layered composite material consisting of a commercial purity titanium (hereafter, CP-Ti) with texture and an isotropic polycrystalline Ti-15V-3Cr-3Sn-3A1 (hereafter, Ti-15-3) alloy, three-dimensional finite element (FE) analysis with an equal strain hypothesis is employed. The validity of the present approach is verified by comparing the FE prediction with the tensile test. The results show that the yield stress and tensile stress of multi-layered composite material can be successfully predicted by FE analysis with the deviation smaller than 5%, and the uniform elongation of multi-layered composite material can be well predicted with the deviation smaller than 1%. The FE analysis also shows that the tension stress normal to interface between components is introduced at the onset of localized necking of multi-layered composites. [doi:10.2320/matertrans.M2010211]

(Received June 18, 2010; Accepted September 9, 2010; Published October 20, 2010)

Keywords: commercial purity titanium/titanium-15-3 multi-layered composite, hot rolled bonding, nominal stress-strain curve, finite element analysis

1. Introduction

High strength steels have been considerably focused especially in the field of automobile to decrease the body weight and meet the requirements of fuel saving. Meanwhile the safety issue of automobiles is also emphasized to pursue a harmony between the social requirement and the natural environment. Generally, high strength can be in contra-diction with the pursuit of high ductility and deformability. In order to solve this problem, the multi-layered structure design of steel sheets is proposed to be an effective approach.1–3) Recently, the uniform elongation of as-quenched martensitic steel was increased from 6% to larger than 50% by the multi-layered structure design through bonding the martensitic steel with the austenitic steel,2) which enhances the expectation to fabricate high-strength structural materials.

Multi-layered composite materials (LCM) consisting of alternating metals or alloys have been widely developed to improve the mechanical properties.1–10) Therefore, the prediction of mechanical properties including the nominal S-S curve of multi-layered composite materials becomes increasingly important in application of multi-layered com-posites. It was reported that the tensile strength of multi-layered composites can be predicted by the average rule.4,5) However, the uniform elongation can not be predicted by the average rule.11)This happens to be the principal purpose of the present paper.

The shape change of specimen and the nominal S-S curves of monolithic Ti-15-3 alloy and CP-Ti have been successful predicted using FE analysis by the present authors.12,13)In the present work, the nominal S-S curves of as-rolled and as-aged multi-layered composite, which is hot-rolled by bonding an anisotropic CP-Ti and an isotropic Ti-15-3 component, are predicted by FE analysis. Discussions were also made on the effect of aging treatment on the nominal S-S curves of the multi-layered composites.

2. Materials and Experimental

2.1 Material

CP-Ti (A component, hereafter) and Ti-15-3 alloy (C component, hereafter) are chosen as component materials in the present study and the chemical composition is listed in Table 1. The CP-Ti was received as a 1 mm thick sheet and then was initially cut into 60mm102mm rectangular piece. As-annealed Ti-15-3 alloy sheets of 5.8 mm thick were ground to 4, 3, 2, 1 mm in thickness and then cut into a rectangular size of 60mm102mm. The multi-layered composites are designed from the aspects of the layer number and the thickness ratio of components. The layer number is chosen as 5-, 7-, and 9-layer, and the thickness ratio of monolithic C to A component is chosen as 1, 2, 3, 4, and 5.8 considering 5-layer design.

2.2 Hot rolled bonding

Hot rolled bonding (HRB) is chosen in order to obtain high bonding quality. Figure 1 shows HRB process of the multi-layered composites. Before stacking, the surfaces of the sheets were degreased and then mechanically ground to achieve good bonding. Afterwards, the sheets were stacked together in an alternating sequence with A and C component such as ACACA for a 5-layered composite, and then fixed at the corners using welding. The stacked sheets were wrapped with stainless steel foils and then evacuated to create a vacuum inside. Surface preparation, stacking and packing stated above were conducted in an argon atmosphere in a vacuum type glove box (type: MDB-1B-0).

The multi-layered composites were kept for 1 h at the temperature of 1123 K, and then hot-rolled to approximately 2.1 mm in thickness with a 2-high’300350mm hot rolling mill at a speed of 10 mmin1. After every pass of rolling,

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Taking HRB5-A1C3 in Table 2 as an example, the number of 5 refers to the layer number of multi-layered composite, and A1C3 means A is 1 mm and C is 3 mm in thickness with an alternating sequence ACACA. It should be noted that the volume fraction of CP-Ti in Table 2 is calculated according to the average layer thickness after bonding.

2.3 Tensile test

Tensile test was performed at a constant crosshead speed of 1 mmmin1 (initial strain rate: 4:8104s1) at room temperature using a universal testing machine with a parallel length of 35 mm and a width of 6.0 mm.

True stress-strain (S-S) relation can be obtained by the nominal stress-strain curve according to constant volume principle as follows:

¼nð1þ"nÞ; "¼lnð1þ"nÞ ð1Þ

whereand"are true stress and true strain, respectively;n and "n are nominal stress and nominal strain, respectively. However, eq. (1) is valid only when the uniform deformation is assumed along the whole specimen. In other words, complete homogeneous deformation of specimen is assumed in eq. (1).

Pino Kocet al.suggested that the fitting accuracy is most important to fit the flow curves because there is no essential difference among Hollomon’s, Swift’s and Ludwig’s law in

describing the flow curves of materials.14) The true S-S relation ¼848:2"0:03 (R2¼0:958) has been successfully

used to predict the nominal S-S curve of as-annealed Ti-15-3 alloy,12) and ¼382:"þ0:004Þ0:10 (R2 ¼0:996) for

as-annealed CP-Ti in transverse direction (TD).13)The relation is obtained by fitting the true S-S curve calculated by eq. (1) according to empirical data until the maximum load point for as-aged Ti-15-3 alloy as follows:

¼1237:1ð"þ0:00072Þ0:038ðR2 ¼0:999Þ ð2Þ

The tensile test of multi-layered composites was inter-rupted at every 5% in elongation based on the indication of extensometer to measure the width, thickness at the end of deformed specimen, and the thickness and width data at each interrupted strain is used as one empirical data to control the displacement of the nodes on both ends of sample for the simulated case in FE prediction. Certainly, prediction results are dependent on the amount of interrupted test.

2.4 Microstructure

Automated electron backscattered diffraction (EBSD) measurement was performed for monolithic as-annealed and as-aged CP-Ti, Ti-15-3 sheets and their multi-layered composites, which was mechanically ground and then electrochemically polished in a solution consisting of acetic acid and perchloric acid. A Carl-Zeiss LEO-1550 type field-emission gun scanning electron microscope, equipped with the orientation imaging microscope (OIM) system, TSL Inc., was applied for the microstructure characterization.

2.4.1 Monolithic CP-Ti and Ti-15-3 alloy

[image:2.595.43.544.84.126.2]

The microstructure and inverse pole figure (IPF) of as-annealed CP-Ti and Ti-15-3 monolithic component are shown in Fig. 2.12,13) The microstructure of as-annealed CP-Ti is composed of equiaxed grains and (0001) crystal planes parallel to normal direction (ND) section as shown in Fig. 2(a). Thus, the anisotropic plasticity of CP-Ti can be assumed in the FE analysis.13)Figure 2(b) shows the micro-structure and IPF (ND section) of as-annealed Ti-15-3 alloy. The IPF reveals that there is no obvious texture and an isotropic deformation can therefore be assumed in the FE analysis.13)

Table 1 Chemical composition of component material (mass%).

Material C H N Fe O Al Cr Sn V Ti

CP-Ti (A) 0.005 0.0022 0.002 0.035 0.035 — — — — Bal.

Ti-15-3 (C) 0.008 0.0086 0.005 0.062 0.104 2.84 3.02 2.93 14.4 Bal.

Surface preparation

Stacking and Fixing

Packing and then Reheating A

C A C A

Multi-layered composite (Aging treatment) HRB

[image:2.595.78.517.87.254.2]

Fig. 1 Schematic illustration of hot rolled bonding (HRB) process (5-layered material).

Table 2 Volume fraction of CP-Ti, total thickness and reduction of multi-layered composites.

Composite Layer number

Volume fraction of CP-Ti (%)

Initial thickness

(mm)

Final thickness

(mm)

Total reduction

(%)

HRB5-A1C1 5 56.3 5 2.1 58

HRB7-A1C1 7 52.4 7 2.1 70

HRB9-A1C1 9 51.7 9 2.1 77

HRB5-A1C2 5 38.5 7 2.1 70

HRB5-A1C3 5 29.5 9 2.1 77

HRB5-A1C4 5 25.0 11 2.1 81

[image:2.595.46.291.323.444.2]
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2.4.2 Multi-layered composite materials

Figure 3(a) and (b) respectively shows the EBSD micro-structure and the IPF of components in multi-layered composite (HRB9-A1C1). The distribution of alloy elements across the interface was plotted as line scans using EDS as shown in Fig. 3(c).

From the Fig. 3(a), it can be found that the microstructure of CP-Ti component is composed of equiaxedgrain with an average grain size of about 50mm, which is close to that of monolithic sample, and the texture shown in Fig. 3(b) is same as that of monolithic as shown in Fig. 2(a). The micro-structure of Ti-15-3 component in HRB9-A1C1 is composed of equiaxed grains with an average grain size of about 85mmas shown in Fig. 3(a) and there is no obvious texture (see Fig. 3(b)). From the Fig. 3(c), the diffusion widths of Ti and V elements are approximately 7.5mm. Further, the results indicate that the diffusion widths of alloy elements are found in the range of 4.0–8.0mmand 5.0–10.0mm respec-tively in as-rolled and as-aged multi-layered composite samples, and the diffusion zone seems to be homogeneous in both thickness and length positions.

2.5 Hardness measurement

Vickers hardness (HV) test was performed for the multi-layered composites using an HV-114 hardness testing machine. The load force is 200 gf for CP-Ti and 1000 gf for Ti-15-3 component and the load time is chosen as 15 s. Figure 4(a) and (b) shows the average hardness of compo-nents in as-rolled and as-aged HRB9-A1C1 materials, respectively. The average hardness of as-annealed CP-Ti and Ti-15-3 is approximately 105 and 255Hv, and the average one of as-aged Ti-15-3 alloy is about 320Hv, which is also shown in Fig. 4 as dot lines. The results indicated monolithic and components in multi-layered composites are very close in hardness.

3. FE Analysis

From Fig. 2, Fig. 3 and Fig. 4, the texture and hardness of monolithic samples are quite close to that of components in multi-layered samples, so, in the present paper, the true stress-strain relations (¼848:2"0:03 for Ti-15-3 and ¼ 382:0ð"þ0:004Þ0:10 for as-annealed CP-Ti in transverse

(a)

IPF ND

(b)

IPF ND

Fig. 2 Microstructure and inverse pole figure of the as-annealed monolithic CP-Ti and Ti-15-3 component (a) CP-Ti; (b) Ti-15-3.

Ti-15-3

ND

CP-Ti

ND

(b)

(a)

ND

RD

A-1

C-1

A-2

C-2

A-3

C-3

A-2

A-3

A-4

A-5 C-1

C-2

C-3

C-4 A-1 Surface

Surface

(c)

0 200 400 600 800 1000 1200 0

20 40 60 80 100

Ti V Cr Al Sn C A C A C

Element Content, wt%

Distance, µm

Distance, µm HRB9-A1C1 A

3800 385 390 395

20 40 60 80 100

Ti V Cr Al Sn

Element Content, wt%

Interface

C A

7.5um

[image:3.595.139.460.73.233.2] [image:3.595.112.487.281.484.2]
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direction (TD)), the material parameters including Poisson ratio (0.33 for Ti-15-3 and 0.3 for CP-Ti), yield stress (720 MPa for Ti-15-3 and 220 MPa for CP-Ti) and Young’s modulus (76.5 GPa for Ti-15-3 and 55 GPa for CP-Ti) obtained by tensile test of monolithic Ti-15-312) and CP-Ti13) are employed in FE prediction. In addition, the true stress-strain relation (eq. (2)) of as-aged Ti-15-3 alloy is also used to predict the nominal S-S curves of as-aged multi-layered composites.

Commercial MSC.MARC 2005 software is used to predict the stress-strain curves in the present work. 7-type element, an eight-node hexahedral element implemented in MSC.MARC 2005, suited to large strain analysis with high accuracy, is chosen in FE prediction. Figure 5 gives a half model in width and the initial mesh of tensile sample (HRB9-A1C1) in FE analysis. The 3-dimensional finite element method is used and the mesh in the length x direction is biased with the fine mesh close to the center of the sample in order to correctly describe stress and strain gradients

expected in the necking zone. The mesh system has 6 and 18 divisions in the y- and z-directions, respectively. There are 8820 three-dimensional elements and 11130 nodes in the model of HRB9-A1C1. The meshes for other specimens with different lengths are in general similar to those shown in Fig. 5. The nodes of interface are shared by two components that is actually reasonable until the delamination occurs, which is not under consideration in the present FE analysis.

The boundary conditions adopted are imposed by control-ling nodes. Two boundary conditions, including a simulated case assuming grip constraint derived from the empirical data in y- and z-direction at both ends of the parallel part of specimen and a free case assuming no y- and z-direction constraint at both ends of the parallel part of sample, are analyzed. In the simulated case, the nodes on both ends of sample (x¼0 and x¼L0¼35mm section) have been

controlled based on empirical data obtained by interrupted test alongy- andz-direction, and the displacement of nodes onx¼L0 section is controlled according to the cross-head

speed.

The true stress-strain relations, the average layer thick-ness, finite element and node number of multi-layered samples used in FE analysis have been listed in Table 3. As stated above, according to the microstructure and the corresponding inverse pole figure as shown in Fig. 2, Mises yield function considering the plastic isotropy are used for Ti-15-3 material, and Hill’s yield function (in 1948) with an anisotropy plasticity model are used for CP-Ti material in the commercial MSC.MARC 2005 software. Hill’s yield function is an extension of the conventional Mises yield function to allow anisotropic material behavior. The function is written as15)

fðijÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

FðyzÞ2þGðzxÞ2þHðxyÞ2þ2L2

yzþ2M2zxþ2N2xy

q

ð3Þ

where F, G, H, L, M and N are constants of plastic anisotropy at different directions. In MSC.MARC, Hill’s yield function eq. (3) is defined by user input consisting of different ratios of yield stress in different directions with respect to a reference stress. In the FE used in this paper, x-axis is the tensile axis,y-axis represents the width direction andz-axis is the thickness direction (see Fig. 5). The yield

stress in thex-axis is taken as the reference stress. Therefore, the yield stress ratio in thex-axis isrx¼x=x¼1:0. The

yield stress ratios in y- and z-directions, ry and rz, have

been changed to reflect different material anisotropy. When the yield stress ratio has been obtained, MSC.MARC will calculate the anisotropy parameters according to the follow-ing equations:

0 50 100 150 200 250 300 350 400 As-rolled

As-annealed Ti-15-3 As-annealed CP-Ti

CP-Ti

CP-Ti

CP-Ti

CP-Ti Ti-15-3

Ti-15-3

Ti-15-3

Ti-15-3

CP-Ti

Hardness,HV

Component

(a)

0 50 100 150 200 250 300 350 400 CP-Ti

CP-Ti

CP-Ti

CP-Ti Ti-15-3

Ti-15-3

Ti-15-3

Ti-15-3 CP-Ti

Hardness,HV

Component

As-aged

As-aged Ti-15-3

As-annealed CP-Ti

(b)

Fig. 4 The average hardness of components in HRB9-A1C1: (a) as-rolled; (b) as-aged.

E

F C

D Mesh Model

C D E

F

x y

x y

z y

L0

[image:4.595.130.466.71.214.2] [image:4.595.54.280.266.368.2]
(5)

F¼1

2 1þ

1

r2

z

1

r2

y

!

G¼1

2 1

r2

z

þ 1

r2

y

1

!

H¼1

2 1þ

1

r2

y

1

r2

z

!

L¼ 3

2r2

xz

M ¼ 3

2r2

yz

N¼ 3

2r2

yx

ð4Þ

where rx¼rz¼1, ry¼0:78, rxy¼ryz¼1:54, rzx¼1:73

for TD.

In the present paper, the yield stress in x-direction is considered the same as that in z-direction according to the texture characteristic of CP-Ti.13)Lagrangian spin for large plastic deformation is adopted in FE analysis, and FE prediction is performed neglecting the occurrence of temper-ature rise due to deformation heating.

4. Results and Discussion

4.1 The nominal stress-strain curve of multi-layered composites

The onset of localized necking has been defined as the termination of uniform deformation by the authors of papers.12,13) The free case assumes uniform deformation during tension without restrict in thickness and width at the ends of specimen, and in the simulated case, non-uniform deformation would occur after uniform deformation due to the whole restraint at both ends of specimen. Therefore, 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.

The predicted nominal stress-strain curves and the empirical one are shown in Fig. 6 for HRB5-A1C3 and HRB9-A1C1. The predicted curves in simulated case are well matched with the empirical one in over the strain range investigated. From Fig. 6, it shows that the nominal stress-strain curves predicted in the simulated case deviate from that predicted in the free case at the critical point, which has been defined as the onset of localized necking (Plocal), shown by an open rectangular on the curves in Fig. 6. The results also indicate that Plocal is fairly larger than the

maximum load point strain, and this means that the uniform deformation occurs fairly far further after Pmax for multi-layered composites.

Since the delamination between CP-Ti and Ti-15-3 components of multi-layered composites is not taken into account in the FE analysis, the second deviation may imply the onset of delamination between components of multi-layered composites. Certainly, for more accuracy prediction of nominal S-S curves, the delamination phenomena must be considered in FE analysis of whole flow curves of multi-layered composites.

It is pointed out that the stress gap is found between the empirical data and the predicted ones. Table 4 lists the stress gap of maximum load between empirical data and FE prediction as well as the average gap of hardness between monolithic and components in multi-layered composites.

Hv(A) and Hv(C) is the hardness gap of multi-layered minus monolithic respectively for CP-Ti and Ti-15-3; for stress gap ¼pe, p, e means predicted and empirical stress, respectively.

From Table 4, it can be seen that there is a deviation between the stress predicted by FE analysis and the empirical data. Apart from hardness gap between monolithic and components in multi-layered composites, the stress gap between empirical data and FE prediction may be associated with residual stress at the interface11,16) and local uneven interface (Fig. 3(a)) instead of even interface assumption in FE analysis (see Fig. 5).

4.2 Strength and uniform elongation of multi-layered composites

Figure 7(a) and (b) compares yield stress (0:2) and tensile

stress (b) predicted by FE analysis and the empirical data. The stresses calculated by the average rule are also shown in Fig. 7(a) and (b), where the straight lines represent the average rule. The uniform elongation"u(UE) is predicted by the maximum-load condition based on the average rule of stress:11)

"u¼

FA"AuþFC"Cu

FTotal

ð5Þ

Table 3 Material parameters and simulated details used in FEM analysis.

Material Element number Node number Average layer thickness, mm

HRB5-A1C1 As-rolled 6300 8162 0.39/0.45/0.38/0.45/0.39

As-aged

HRB7-A1C1 As-rolled 8820 11130 0.28/0.34/0.26/0.30/0.26/0.34/0.28

As-aged

HRB9-A1C1 As-rolled 11340 14098 0.25/0.36/0.22/0.15/0.15/0.15/0.22/0.36/0.25

As-aged

HRB5-A1C2 As-rolled 8820 11130 0.27/0.64/0.26/0.64/0.27

As-aged

HRB5-A1C3 As-rolled 8820 11130 0.21/0.74/0.20/0.74/0.21

As-aged

HRB5-A1C4 As-rolled 8820 11130 0.18/0.78/0.16/0.78/0.18

As-aged

HRB5-A1C5.8 As-rolled 8820 11130 0.13/0.85/0.11/0.85/0.13

[image:5.595.41.547.84.269.2]
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where the uniform elongation of each component and the load supported by each layer are denoted, respectively, by"

u andF, and ‘A’ and ‘C’ are substituted for the asterisks to indicate the CP-Ti and Ti-15-3 component, respectively, while Ftotal is the total load supported by the multi-layered composite. Figure 7(c) and (d) compares the uniform elongation predicted by FE analysis and the empirical data.

From Fig. 7(a) and (b), it can be found that the yield stress and tensile stress are consistent with those predicted by FE and by average rule of both as-rolled and as-aged multi-layered composites with a relative error smaller than 5%. It is interesting that FE analysis can well predict the uniform elongation for both as-rolled and as-aged multi-layered composites with a deviation smaller than 1%, but the aver-age rule fails to predict uniform elongation especially for the as-rolled multi-layered composites (see Fig. 7(c) and (d)).

4.3 Normal stress at interface by FE analysis

The maximum normal stress at interface between middle layers as a function of nominal strain for as-rolled multi-layered composites is shown in Fig. 8. From the Fig. 8(a), it can be found that the normal stress at interface keeps nothing until the critical nominal strain, and then rapidly increases with an increase in strain. The critical nominal strain gradually decreases from 0.17 to 0.12 with an increase in volume fraction of Ti-15-3 component from 43.7% to 82.1% for 5-layer materials. After the critical strain, the normal stress remarkably increases. In addition, the critical nominal strain is shown as 0.17, 0.16 and 0.16 respectively for HRB5-A1C1, HRB7-A1C1 and HRB9-A1C1 materials as shown in Fig. 8(b).

[image:6.595.120.474.74.365.2]

The critical nominal strain predicted by FE analysis shown in Fig. 8 as a function of Plocal of multi-layered composites is shown in Fig. 9. It can be seen that the critical nominal strain is consistent with the Plocal strain. Accord-ingly, it can be concluded that the onset of localized necking becomes the origin of the normal stress at interface as shown in Fig. 8.

Table 4 Hardness gap between monolithic and components in multi-layered composites and stress gap at maximum load between FE prediction and empirical data.

Material Condition CP-Ti

Hv(A)

Ti-15-3

Hv(C)

Stress gap between FE prediction

and empirical data,

/MPa

HRB5-A1C1 As-rolled +2 +13 13

As-aged 2 +4 +2

HRB7-A1C1 As-rolled +3 +7 10

As-aged 1 +10 +15

HRB9-A1C1 As-rolled +1 +10 13

As-aged 4 +9 +9

HRB5-A1C2 As-rolled +6 12 25

As-aged 6 +11 +19

HRB5-A1C3 As-rolled +1 11 18

As-aged 3 +2 +2

HRB5-A1C4 As-rolled +7 7 25

As-aged 1 +15 +32

HRB5-A1C5.8 As-rolled 3 5 20

As-aged 1 +4 +15

0.00 0.05 0.10 0.15

0 100 200 300 400 500 600 700 800 900 1000

Plocal

Nominal stress (MPa)

Nominal strain Free case Simulated case Empirical data Pmax

As-aged HRB5-A1C3 (b)

0.000 0.05 0.10 0.15 0.20 0.25

100 200 300 400 500 600 700

Plocal

Nominal stress (MPa)

Nominal strain Free case Simulated case Empirical data Pmax

As-rolled HRB5-A1C3 (a)

0.000 0.05 0.10 0.15 0.20 0.25 0.30

100 200 300 400 500 600 700

Plocal

Nominal stress (MPa)

Nominal strain Free case Simulated case Empirical data Pmax

As-rolled HRB9-A1C1 (c)

0.000 0.05 0.10 0.15

100 200 300 400 500 600 700 800 900 1000

Plocal

Nominal stress (MPa)

Nominal strain Free case Simulated case Empirical data

Pmax

As-aged HRB9-A1C1 (d)

[image:6.595.47.290.455.666.2]
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[image:7.595.123.475.74.355.2]

Figure 10 schematically shows a tensile test section of 5-layered composites consisted of a soft CP-Ti and a hard Ti-15-3 component, where m is the flow stress of 5-layered

composite;r is the instantaneous value of the uniaxial flow stress in the hard layer. It is assumed that necking starts in the Ti-15-3 layer (C layer) while the CP-Ti layer (A layer) continues to undergo stable straining. Therefore, CP-Ti layer will impose a constraint to the necking of the Ti-15-3 layer. This constraint is represented in Fig. 10 by a tensile stress

tacting in a direction perpendicular to the component in the necked region. This will tend to raise the effective flow strain and improve the uniform elongation and tensile ductility of Ti-15-3 layer.

However, it is assumed that the bonding has a high quality to discuss the constraint by the tensile stress t acting in a 0.000 0.05 0.10 0.15 0.20 0.25

100 200 300 400 500

HRB5 A1C5.8 A1C4 A1C3 A1C2 A1C1

z x

Node position Center

Normal stress at interface,MPa

Nominal strain

0.000 0.05 0.10 0.15 0.20 0.25 100

200 300 400 500

z x

Node position Center HRB9

Normal stress at interface,MPa

Nominal strain

HRB5-A1C1 HRB7-A1C1 HRB9-A1C1

(a) (b)

Fig. 8 The normal stress at interface between middle layers as a function of nominal strain for multi-layered composites (a) 5-layer materials (b) 5, 7 and 9-layer materials.

0 100 200 300 400 500 600 700 800 9001000 0

100 200 300 400 500 600 700 800 900 1000

1

Yield stress

Empirical data Average rule

Empirical stress

Predicted stress by FE

1

0 100 200 300 400 500 600 700 800 9001000 0

100 200 300 400 500 600 700 800 900 1000

1

Tensile stress

Empirical data Average rule

Empirical stress

Predicted stress by FE

1

0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 14

1 Uniform elongation

(As-rolled)

Empirical data Average rule

Empirical data

Predicted data by FE

1

0 2 4 6 8 10 12 14

0 2 4 6 8 10 12 14

1 Uniform elongation

(As-aged)

Empirical data Average rule

Empirical data

Predicted data by FE

1

Fig. 7 Comparison between empirical data and predicted ones by FE analysis and average rule. (a) yield stress; (b) tensile stress; (c) UE (as-rolled); (d) UE (as-aged).

0.00 0.05 0.10 0.15 0.20

0.00 0.05 0.10 0.15 0.20

As-rolled As-aged

P

local

strain

Critical nominal strain

Fig. 9 The critical nominal strain predicted as a function ofPlocalstrain of

[image:7.595.96.502.402.561.2] [image:7.595.77.260.609.757.2]
(8)

direction perpendicular to the component. In this study, the CP-Ti layer is most important in delaying the localized necking of Ti-15-3 component to improve the strength and uniform deformation in uniaxial tension. For multi-layered composites, the tensile ductility is controlled not only by the strain-hardening rate of components but also by the hardness ratio of components. CP-Ti and Ti-15-3 layers bear the load given by the tensile test together for the same level throughout the testing process and retain the deforma-tion as uniform as possible. This kind of deformadeforma-tion mechanism will be apt not only to destroy the interfacial bonding strength but also to cause the delamination due to necking.

5. Conclusion

(1) The nominal stress-strain curves of multi-layered composites have been well predicted by means of true stress-strain curves of monolithic component with the deviation smaller than 5% in stress, and the uniform elongation of multi-layered composite material can be successfully pre-dicted with the deviation smaller than 1%.

(2) The FE analysis indicates that the normal stress at interface between CP-Ti and Ti-15-3 components keep nothing until a critical nominal strain corresponding to the

Plocal strain, implying that the onset of localized necking becomes the origin of the normal stress at interface of multi-layered composites.

(3) The onset of localized necking of multi-layered composites is fairly larger than the maximum load point strain. The origin of localized necking is concluded not due to the plastic instability but due to the deformation constraint at both ends of multi-layered samples.

Acknowledgment

This study was conducted as part of the LISM (Layer-Integrated Steels and Metals) Project funded by Ministry of Education, Culture, Sports, Science and Technology of Japan.

REFERENCES

1) D. R. Lesuer, C. K. Syn, O. D. Sherby, J. Wadsworth, J. J. Lewandowski and W. H. Hunt Jr: Int. Mat. Rev.41(1996) 169–197. 2) S. Nambu, M. Michiuchi, Y. Ishimoto, K. Asakura, J. Inoue and T.

Koseki: Scr. Mater.60(2009) 221–224.

3) T. Koseki, J. Inoue, T. Suzuki, K. Nagai and Y. Taniguchi: Metal80 (2010) 3–7. (in Japanese)

4) S. L. Semiatin and H. R. Piehler: Metall. Trans. A10(1979) 97–107. 5) Dong Nyung Lee and Yoon Keun Kim: J. Mater. Sci.23(1988) 558–

564.

6) J. Inoue, S. Nambu, Y. Ishimoto and T. Koseki: Scr. Mater.59(2008) 1055–1058.

7) S.-Hoon Choi, K.-Hwan Kim, K. Hwan Oh and D. Nyung Lee: Mater. Sci. Eng. A222(1997) 158–165.

8) K. S. Ravichandran, S. S. Sahay and J. G. Byrne: Scr. Mater.35(1996) 1135–1140.

9) Shyong Lee, J. Wadsworth and Oleg D. Sherby: J. Compos. Mater.25 (1991) 842–853.

10) C.-Yuan Chen, H.-Long Chen and W.-Sing Hwang: Mater. Trans.47 (2006) 1232–1239.

11) S. L. Semiatin and H. R. Piehler: Metall. Trans. A10(1979) 85–96. 12) L. Li, F. Yin and K. Nagai: Mater. Trans. 51(2010) doi:10.2320/

matertrans.M2010033.

13) L. Li, K. Nagai and F. Yin: Mater. Trans. 51(2010) doi:10.2320/ matertrans.M2010094.

14) P. Koc and B. Stok: Comput. Mater. Sci.31(2004) 155–168. 15) Z. L. Zhang, J. Odegard, O. P. Søvik and C. Thaulow: Int. J. Solids

Structures38(2001) 4489–4505.

16) Y. W. Bao, S. B. Su and J. L. Huang: J. Compos. Mater.36(2002) 1769–1778.

σr σm

σt σt

A C A C A

Tensile direction

σt σr

[image:8.595.113.229.78.235.2]

σm

Figure

Table 2Volume fraction of CP-Ti, total thickness and reduction of multi-layered composites.
Fig. 2Microstructure and inverse pole figure of the as-annealed monolithic CP-Ti and Ti-15-3 component (a) CP-Ti; (b) Ti-15-3.
Fig. 4The average hardness of components in HRB9-A1C1: (a) as-rolled; (b) as-aged.
Figure 7(a) and (b) compares yield stress (�The stresses calculated by the average rule are also shownin Fig
+4

References

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