Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal...

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International Journal of Plasticity 21 (2005) 101–117

Anisotropic plastic deformation ofextruded aluminum alloy tube under axial

forces and internal pressure

T. Kuwabara a,*, K. Yoshida a, K. Narihara a, S. Takahashi b

a Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, 2-24-16

Naka-cho, Koganei-shi, Tokyo 184-8588, Japanb Advanced Vehicle Manufacturing and Prototype Technology Department, NISSANMotor Co., Ltd., 560-2

Okatsukoku, Atsugi, Kanagawa, 243-0192, Japan

Received in final revised form 12 March 2004

Available online 19 May 2004

Abstract

The anisotropic plastic deformation behavior of extruded 5000 series aluminum alloy tubes,

A5154-H112, of 76 mm outer diameter and 3.9 mm wall thickness is investigated, using a

servo-controlled tension-internal pressure testing machine. This machine is capable of ap-

plying arbitrary stress or strain paths to a tubular specimen using an electrical, closed-loop

control system. Detailed measurements were made of the initial yield locus, contours of plastic

work for different levels of work-hardening, and the directions of the incremental plastic strain

vectors for both linear and combined stress paths. It is found that the measured work contours

constructed in the principal stress space are similar in shape, and that the directions of the

incremental plastic strain vectors remain almost constant at constant stress ratios. The work-

hardening behavior predicted using Hosford’s or the Yld2000-2d yield functions under the

assumption of isotropic hardening agrees closely with the observations for both linear and

combined stress paths. The material is thus found to work-harden almost isotropically. Both

yield functions are effective phenomenological plasticity models for predicting the anisotropic

plastic deformation behavior of the material.

� 2004 Elsevier Ltd. All rights reserved.

*Corresponding author. Tel.: +81-42-388-7083; fax: +81-42-385-7204.

E-mail addresses: [email protected] (T. Kuwabara), [email protected] (S. Takah-

ashi).

0749-6419/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2004.04.006

mail to: [email protected]

102 T. Kuwabara et al. / International Journal of Plasticity 21 (2005) 101–117

Keywords: A. Cutting and forming; B. Anisotropic material; B. Constitutive behavior; C. Mechanical

testing; Yield function

1. Introduction

Tube hydroforming is the process of forming tubular components under hy-

draulic pressure. This process allows reduction in weight and costs and gives greater

structural strength and rigidity in automotive body structures. The deformation

histories of specimen materials in tube hydroforming are complex, so that it is dif-

ficult for die designing engineers to predict defects in formed parts, including fracture

and springback. Product development cycles have shortened recently, ruling out

traditional empirical approaches for determining the optimum forming conditions.

Consequently, there is a need for accurate simulation techniques for tube hydro-forming using finite element analysis. For accurate and time-effective finite element

simulations, it is vital to use accurate phenomenological plasticity models based on

anisotropic yield functions. For example, the effect of differing anisotropic plasticity

models on sheet necking predictions has recently been studied by Kuroda and

Tvergaard (2000) and Wu et al. (2003). For a detailed review of anisotropic yield

functions, see Banabic (2000).

There have been many experimental studies of multiaxial testing of thin-walled

tubular specimens, involving thin-walled specimens loaded in combined tension-tor-sion or tension-internal pressure modes; see Hecker (1976), Ikegami (1979) and

Michno and Findley (1976) for further references. However, few experimental studies

have looked at the anisotropic behavior of extruded aluminiumalloy tubes, which have

recently been extensively used to produce lightweight automotive body structures.

We study below the anisotropic plastic deformation behavior of a 5000 series

extruded aluminum alloy tube, A5154-H112, using a servo-controlled tension-in-

ternal pressure testing machine. The initial yield locus and contours of plastic work

in stress space, as well as the directions of the incremental plastic strain vectors, aremeasured accurately for linear and combined stress paths. The validity of conven-

tional anisotropic yield functions is verified by comparing the observed data with

theoretical predictions based on yield functions.

2. Experimental methods

2.1. Test material

The test material was a 5154-H112 extruded aluminum alloy tube of 76.3 mm

outer diameter and 3.9 mm wall thickness. The initial gap between chucking jigs of

the testing machine, i.e., the gage length of the specimen, was set to be 160 mm in all

experiments. Fig. 1 shows the uniaxial true stress-uniaxial logarithmic plastic strain

curves measured in the axial ð/Þ and circumferential ðhÞ directions of the specimens.

These curves were generated from stress control tests; see Section 2.3 for details of

0.0 0.1 0.2 0.30

100

200

300

400

σφ = 328{1-0.72exp(-8.35

p

φ)}

Tru

e st

ress

σ φ, σ

θ/M

Pa

Logarithmic plastic strain p

φε ε, p

θ

σφ - p

φ

σθ - p

θε

ε

ε

Fig. 1. Uniaxial stress–strain curves in the axial ð/Þ and circumferential ðhÞ directions of specimens.

T. Kuwabara et al. / International Journal of Plasticity 21 (2005) 101–117 103

the testing procedures. The serrated stress–strain curves observed are typical of 5000series aluminum alloys. The dotted line in the figure is fitted to the data using Voce’s

hardening law (Voce, 1955). The flow stress in the h-direction was larger than that in

the /-direction, so that the flow stress is anisotropic. Table 1 shows the work-

hardening characteristics and r-values, r/ and rh, obtained from the uniaxial tensile

tests. Variation of the r-values with plastic strain is small. We assume, therefore, that

the r-values of the present material are constant irrespective of the amount of strain

in determining the material coefficients of the anisotropic yield functions that are

used in the plasticity analyses (Sections 3 and 4).Fig. 2 shows the wall thickness distribution measured in the circumferential di-

rection of a specimen. Variations in the wall thickness appear to be caused by the

elastic deformation of extruded dies, and are ±1.1% of the mean thickness, 3.87 mm.

Our preliminary measurements confirm that all specimens used in this study had

similar tendency and magnitude of variation in wall thickness.

2.2. Testing machine

Fig. 3 shows a schematic of the combined tension-internal pressure testing ma-

chine used in this work. This testing apparatus was originally designed and built by

Table 1

Mechanical properties of test material

Tensile direction r0:2 (MPa) ca (MPa) na e0a rb

/ 93 522 0.39 0.008 0.36

h 93 536 0.37 0.007 0.59

aApproximated using r ¼ cðe0 þ epÞn for 0:0016 ep 6 0:20.bMeasured at uniaxial plastic strain ep ¼ 0:10–0:20.

Fig. 2. Thickness distribution of a specimen.

Load cellSpecimen

Pantograph-typelink mechanism

T

IntensifierP

Hydraulic cylinder

T

Hydraulic cylinder

Fig. 3. Scheme of experimental apparatus.


Kuwabara et al. (2003). Axial loads were applied to the specimen by opposing hy-

draulic cylinders and internal pressure by an intensifier; both were servo-controlled

using an electrical, closed-loop control system as described below.

The deformation behavior of the specimens was measured using a spherometer

and two strain gages, as shown in Fig. 4. The spherometer consists of a displacement

meter and two wedges. The distance between the wedges was determined to be 46mm, and the radius of curvature, R/, in the axial direction of the bulged specimen

was confirmed in preliminary experiments to be almost constant, i.e., dR/=d/ ¼ 0, in

the measurement section. The spherometer has the freedom to rotate around an axis

vertical to the central axis of the specimen, so that the pointed ends of the dis-

placement meter and the two wedges are always in contact with the bulged surface of

Fig. 4. Set-up of spherometer and strain gages.


the specimen. For real time measurement of R/, a pantograph-type link mechanism,

as proposed by Shiratori and Ikegami (1968), was used so that the center of thespecimen was always positioned at the center of the testing apparatus, where the

spherometer was located. Using thick-walled theory, Jones and Mellor (1967) sug-

gested that the concept of a thin-walled cylinder is an approximation, the accuracy of

which varies with stress ratio, and showed that taking the stress and strain mea-

surements with reference to the central section of the tube wall gives a good ap-

proximation. Therefore, we decided to evaluate the stress- and strain-components of

specimens at the mid wall.

The axial and circumferential membrane strains, e/ and eh, were measured usingstrain gages designed to measure large plastic deformations (Tokyo Sokki Ken-

kyujo Co., YFLA-2). These were mounted on the centerline of the specimen as

shown in Fig. 4. e/ and eh were assumed to be uniform in the circumferential

direction of the specimen, and were evaluated at the mid wall using the following

equations:

e/ ¼ e/S � lnR/

R/ � ðt=2Þ ;

eh ¼ lnD� tD0 � t0

¼ lnD0 expðehSÞ � t

D0 � t0;

ð1Þ

where e/S and ehS are the axial and circumferential true strains measured on the outersurface of the specimen, D0 and t0 are the initial outer diameter and wall thickness of

the specimen, and D and t are the deformed outer diameter and wall thickness

measured at the center of the specimen. The current wall thickness t was determined

under the assumption of constant volume and neglecting elastic strains as

t ¼ t0 expð�e/–ehÞ: ð2Þ
The maximum strain to which the strain gages could maintain contact to a
specimen was 10–15%, varying according to the strain ratio given to the specimen.

Accordingly, we changed strain gages every 10–15% strain measurement.


The axial and circumferential stresses, r/ and rh, were calculated as the values at

the mid wall, using the following equations based on the equilibrium requirements at

the center of the specimen:

Fig. 5

r/ ¼ PpðD=2� tÞ2 þ TpðD� tÞt ;

rh ¼ðR/ � tÞðD� 2tÞ

ð2R/ � tÞ t P � D� t2R/ � t

r/;

ð3Þ

where T and P are measured values of the axial load and the internal pressure,

respectively.

Fig. 5 shows the closed-loop electrical feedback circuit used for controlling the

true stress components, r/ and rh. The measured values of T, P, e/, eh and R/ were

inputs to a personal computer, and the true stress components, r/ and rh, werecalculated from Eq. (3). These values of r/ and rh were compared with the command

values of r/ and rh. Based on the discrepancy, two PI-controllers then output

command signals for controlling T and P every 0.01 s. The command signals for T

and P were then compared with the measured values, and the servo-valves were

controlled according to the difference between the measured and command values of

T and P so as to activate the hydraulic cylinders and intensifier, such that the desired

values of r/ and rh are applied to the specimen. The resolution in the measurements

for the axial loads was 7 N, and that for the internal pressure was 1 kPa. Resolutionin the measurements for strain components was 6 · 10�6. The outputs of T, P, e/, eh,r/, rh and R/ were recorded every 0.3 s with A/D data conversion and a personal

computer, and were saved on a disk.

σθ

σ

−

−

φ

PI-controller

−+

PI-controller

+−

Convertto stress

∆T

Σ

Σ+

+

Tensioncommand

Pressurecommand

Personal computer

D/A

A/D

D/A

Command for σφ

Command for σθ

∆P

∆T

∆P

Strain amp.

Strain amp.

Strain amp.

Strain amp.

Strain amp. Load cell

Strain gage

Strain gage

Spherometer

Pressure gage

Servovalve T

Servoamp.

Servovalve

Hydraulicintensifier

Hydrauliccylinder

PServoamp.

. Closed-loop, electrical feedback circuit used for controlling the true stress components, r/ and rh.


2.3. Experimental procedure

To evaluate the work-hardening behavior of the test material under biaxial ten-

sion, we consider the notion of the contour of plastic work in stress space (Hill and

Hutchinson, 1992; Hill et al., 1994; Barlat et al., 1997). First, a uniaxial tensile test in

the axial direction of the specimen (r/:rh ¼ 1:0) was carried out, and the axial truestress r0 and plastic work W dissipated per unit volume were determined for par-

ticular values of the axial logarithmic plastic strain ep0. Next, biaxial stress tests were

carried out with the stress ratios in certain proportions: r/:rh ¼ 1:0, 4:1, 2:1, 4:3, 1:1,

20:23, 3:4, 1:2, 1:4 and 0:1; here r/:rh ¼ 20:23 is the stress ratio at which the ratio of

the incremental plastic strain components, dep/:deph , is close to unity. Finally, groups

of stress points, ðr/; rhÞ, for which the same amount of plastic work asW was needed

were plotted in principal stress space so as to construct contours of plastic work

corresponding to particular values of ep0. When ep0 is small enough, the correspondingwork contour is effectively a yield locus of the material. In the present study, all

specimens were loaded so that the rate of change of von Mises’s equivalent stress was

constant at 0.2–0.35 MPa/s. Strain rate control test was difficult to perform because

of the serrated flow of the test material.

We have not used the offset plastic strain in the definition of a yield locus, in

contrast to many authors. Since the tested materials are anisotropic, the plastic work

definition of yielding, which retains definite physical meaning, is more appropriate

than the offset plastic strain definition, which is necessarily the same only for vonMises-type materials.

3. Experimental results

3.1. Measured contours of plastic work

Fig. 6 shows the experimental data points making up contours of plastic work fordifferent levels of ep0; values of the stress components are normalized by r0 corre-

sponding to each ep0. Two specimens were tested for each stress ratio, and each data

point in Fig. 6 represents the mean value. The experimental scatter was 8% of the

flow stress for r/:rh ¼ 1:2, and not more than 2–3% of the flow stress at other stress

ratios. The normalized data points mostly lie in a very narrow region at all stress

ratios. This means that successive work contours for different ep0 are similar in shape,

so that the material work-hardens almost isotropically. The work contours bulge

sharply in the directions of r/:rh ¼ 20:23–3:4.Also depicted in Fig. 6 are the yield loci based on an isotropic yield function of

von Mises (1913), Hill’s quadratic yield function (Hill, 1948), and higher-order

yield functions of Hosford (Hosford, 1979; Logan and Hosford, 1980) and Barlat

and Lian (1989). The material coefficients included in Hill’s quadratic, Hosford’s

and Barlat–Lian’s yield functions were determined using r0, r/ and rh. The ex-

ponent of the stress terms in Hosford’s and Barlat–Lian’s yield functions was se-

lected to be eight in order to represent the behavior of fcc materials, as

Fig. 6. Contours of plastic work compared with theoretical yield loci. Different markers correspond to

contours of plastic work for different values of ep0 , the axial logarithmic plastic strain attained in the

uniaxial tensile test of a specimen. The stress values for a particular value of ep0 are normalized by the

uniaxial tensile true stress, r0, that corresponds to ep0 .


recommended by Hosford (1979) and Logan and Hosford (1980). Hill’s quadratic

yield function has been widely used because of its mathematical simplicity, even

thought it fails to predict the anomalous behaviour of some materials with average

r-value less than unity (Woodthorpe and Pearce, 1970). Hosford’s yield function is

used here for comparison with the work contours in view of its analytical sim-

plicity. Another reason is that Hosford’s yield function is able to closely represent

the yield loci, work-hardening behavior and the directions of the incremental

plastic strain vectors measured in the biaxial tension tests of steel sheets withdifferent r-values (Kuwabara and Ikeda, 2002; Kuwabara et al., 1998; Kuwabara

et al., 2002), and also low carbon steel tube (Kuwabara et al., 2003) and 6000

series aluminum alloy sheet (Kuwabara and Van Bael, 1999). However, since

Hosford’s yield function does not include the shear stress component, it can de-

scribe only planar isotropy or cases in which the principal stress and anisotropy

axes are parallel. In contrast, Barlat–Lian’s yield function is able to describe the

behavior of orthotropic sheet metals under a general plane stress state (two normal

stresses and one shear stress).Fig. 6 shows that the yield locus based on Hosford’s yield function is in better

agreement with the observed work contours than the other yield functions. However,

Hosford’s yield function is able still not to reproduce the bulging of the observed

work contours in the directions of r/:rh ¼ 20:23–3:4.

Fig. 7 shows the observed work contours compared with the theoretical yield

loci based on the Yld2000-2d yield function (Barlat et al. (2003) and Yoon et al.

(2004); hereafter referred to Yld2000), as well as those calculated using Hosford’s

Fig. 7. Contours of plastic work compared with theoretical yield loci based on (a) Hosford’s and Yld2000-

2d yield functions and (b) Yld2000-2d yield functions. The short lines attached to the data points in (a)

represent the directions of the incremental plastic strain vectors. The material coefficients of the Yld2000

depicted in (a) were determined using r0, r90, rb, r/, rh and rb, and those of the Yld2000 in (b) were

determined using r0, r90, r/ and rh, as well as the true stress components and the ratio of incremental

plastic strain components measured at the stress ratio, r/:rh ¼ 20:23.



yield function for comparison (Fig. 7(a)). The short lines attached to the data

points in Fig. 7(a) represent the directions of the instantaneous incremental

plastic strain vectors, determined from the corresponding biaxial stress–strain

curves. The material coefficients of the Yld2000 depicted in Fig. 7(a) were de-

termined using r0, r90, rb, r/, rh and rb, where r90 is the uniaxial true stress in

the circumferential direction of the specimen, rb and rb are, respectively, the truestress and the ratio of incremental plastic strain components, deph=de

p/, measured

for equibiaxial tension, i.e., r/:rh ¼ 1:1. The exponent of the stress terms in the

Yld2000 yield function was selected to be eight. The Yld2000 appears to be

superior to Hosford’s yield function in reproducing the bulge in the work con-

tours for directions of r/:rh ¼ 20:23–3:4. In particular, the yield loci based on

Yld2000 are in perfect agreement with the work contours for ep0 ¼ 0:002, 0.01

and 0.025.

To improve the degree of coincidence of Yld2000 with the experimental workcontours, the material coefficients of Yld2000 were recalculated using the true

stress components and the ratio of incremental plastic strain components mea-

sured at the stress ratio, r/:rh ¼ 20:23, as well as the uniaxial tension test data, r0,

r90, r/ and rh. The yield loci determined using the recalculated Yld2000 are shown

as solid lines in Fig. 7(b). It is clear that the new yield loci based on the recal-

culated Yld2000 are in much better agreement with the work contours than those

in Fig. 7(a), representing the bulge in the work contours in the directions of

r/:rh ¼ 20:23–3:4.

3.2. Measured directions of incremental plastic strain vectors

Fig. 8 compares directions of the incremental plastic strain vectors, b, measured at

different values of ep0, with the directions of the local outward normals to the theo-

retical yield loci. The material coefficients of the Yld2000 used in the calculation were

determined using the true stress components and the ratio of incremental plastic

strain components measured at the stress ratio, r/:rh ¼ 20:23, as well as r0, r90, r/and rh, for e

p0 ¼ 0:15 (see Fig. 7(b)). In the figure, the stress ratio is represented as the

direction of the stress path, u, in stress space. The angles b and u are taken to be zero

in the direction of r/-axis and positive for anti-clockwise rotation. Two specimens

were tested for each stress ratio, and the maximum difference in the angle b observed

for the specimens between the different plastic work contours was 6� for any given

stress ratio. This means that, at least for the measured range of the work contours

shown in Fig. 7, the directions of the incremental plastic strain vectors remained

almost constant when the stress ratio was constant. Also, the observed values of bwere 27± 2� for equibiaxial tension, u ¼ 45�, implying strong anisotropy of the

material tested. It was found that b was close to 45� for r/:rh ¼ 20:23 (in fact

u ¼ 49�).Again, the predictions based on Hosford’s and the Yld2000 yield functions are in

good agreement with the experimental results. The maximum difference of the di-

rections of the incremental plastic strain vectors from directions of the local outward

normals for Hosford’s yield locus was 10�, and for Yld2000 was 7�.

Fig. 8. Directions of the incremental plastic strain vectors, b, compared with directions of the local

outward normals to theoretical yield loci.


4. Discussion

Figs. 6–8 show that the measured work contours constructed in the principal

stress space are similar in shape, and that the directions of the incremental plastic

strain vectors remained almost constant at constant stress ratio. Moreover, we have

confirmed that these observations agree closely with theoretical predictions based on

Hosford’s and the Yld2000 yield functions. We therefore conclude that the alumi-

num alloy tubes used in the present work work-hardens almost isotropically, andthat Hosford’s and the Yld2000 yield functions can be regarded as instantaneous

plastic potentials for the material, at least for linear stress paths. It follows that we

can predict the anisotropic plastic deformation behavior of the material by using

Hosford’s or Yld2000 yield functions as phenomenological plasticity models with the

assumption of isotropic hardening.

To verify this assertion, the equivalent stress–equivalent plastic strain curves

were calculated from the true stress–logarithmic plastic strain curves, r/–ep/ and

rh–eph , observed for all stress ratios, using both the Yld2000 and Hosford’s yield

functions. The material coefficients of the Yld2000 used in the calculation were

determined using the flow stresses and the ratio of incremental plastic strain

components measured at the stress ratio, r/:rh ¼ 20:23, as well as the uniaxial

tension test data, r0, r90, r/ and rh, for ep0 ¼ 0:15 (see Fig. 7(b)). Fig. 9 shows the

results calculated using Yld2000. The calculated equivalent stress–equivalent plastic

strain curves fall almost on a single curve irrespective of the stress ratio. Although

the data is not included in Fig. 9, we verified that Hosford’s yield function gave

almost the same result as Yld2000. These yield functions therefore provide effective

Fig. 9. Equivalent stress–equivalent plastic strain curves calculated for various linear stress paths. The

yield function used is the Yld2000-2d, in which the material coefficients were determined using the flow

stresses and the ratio of incremental plastic strain components measured at the stress ratio, r/:rh ¼ 20:23,

as well as r0, r90, r/ and rh for ep0 ¼ 0:15; see in Fig. 7(b).


phenomenological plasticity models for predicting the anisotropic plastic defor-

mation behavior of the material used in the present work, at least for linear stress

paths.

In tube hydroforming, tubular materials are often subjected to prebending

followed by pressurization in the cavity of female dies. In the prebending process,

material elements located at the convex side of the bent portion of the tube are

first subjected to near plane-strain tension in the axial direction, and then to nearplane-strain tension in the circumferential direction by the pressurization process.

It is therefore of interest, from an industrial viewpoint, to check experimentally

whether the assumption of isotropic hardening of the present material still holds

in such combined loading. To verify the assumption, combined loading experi-

ments were carried out as follows, and the resulting equivalent stress–equivalent

plastic strain curves were observed. In the first stage of loading, specimens were

subjected to axial tension under constant stress ratio, r/:rh ¼ 2:1, until ep/ reached

0.04, 0.07, 0.093 and 0.12. After unloading, as the second step, the specimenswere subjected to circumferential expansion under constant stress ratio,

r/:rh ¼ 1:2, until they burst. Fig. 10 shows the equivalent stress–equivalent plastic

strain curves calculated for the combined loadings described above using the

Yld2000 yield function. The Yld2000 used here is the same as that used in Fig. 9.

Again, the calculated equivalent stress–equivalent plastic strain curves all lie close

to a single curve irrespective of the amount of prestrain, ep/. It follows that we

may still assume that the material work-hardens isotropically for this type of

combined loading. We conclude that the Yld2000 yield function is still an

Fig. 10. Equivalent stress–equivalent plastic strain curves calculated for the combined loadings typically

observed in tube hydroforming: near plane-strain tension in the axial direction followed by near plane-

strain tension in the circumferential direction. The same Yld2000-2d is used in the calculation as in Fig. 9.


effective phenomenological plasticity model under the assumption of isotropic

hardening, and is useful for predicting the anisotropic plastic deformation be-

havior of the material under the combined loadings typically observed in tube

hydroforming.

Since some material elements of tubular specimens are subjected to complex

stress/strain paths in tube hydroforming, it is worth checking whether the isotropic

hardening assumption remains valid for non-linear stress paths. To verify the as-sumption, the following bilinear stress paths, A and B, were applied to specimens,

and the resulting strain paths were observed.

Stress path A: The specimen was first subjected to a linear stress path with the stressrate ratio constant as _r/: _rh ¼ 20:23 until the stress reached ðr/; rhÞ ¼ ð160; 184Þ(MPa), and then the stress rate ratio was changed to _r/: _rh ¼ 0:1 and held constant.

Stress path B: The specimen was first subjected to a linear stress path with the

stress rate ratio constant as _r/: _rh ¼ 0:1 until the stress reached ðr/; rhÞ ¼ð0; 136Þ (MPa), and then the stress rate ratio was changed to _r/: _rh ¼ 20:23 and

held constant.

In both cases the loadings were terminated at the same stress point

ðr/; rhÞ ¼ ð160; 320Þ (MPa). Fig. 11 shows the observed strain paths for both stress

paths, compared with the results calculated using conventional yield functions under

the assumption of isotropic hardening. Again, the strain paths predicted using

Yld2000 are in closest agreement with the observed ones. We conclude that the

Fig. 11. Strain paths observed for bilinear stress paths, A and B, compared with the paths predicted using

several theoretical yield functions. The same Yld2000-2d is used in the calculation as in Fig. 9.


isotropic hardening assumption remains valid, at least for the bilinear stress paths in

Fig. 11.All plasticity analyses carried out in this work are based on the normality flow

rule. However, this rule does not correctly represent the plastic deformation be-

havior of metals when large deviations from proportional loading occur. Recently,

Kuroda and Tvergaard (1999) proposed the use of an abrupt strain path change in

order to determine the shape of the subsequent yield surface in the vicinity of the

instantaneous loading point. When this method is used for polycrystal plasticity,

based on the Taylor model for either fcc or bcc crystal structure, it clearly shows

the existence of a vertex at the current point of loading. Furthermore, when thestress point moves in a direction nearly tangential to the apparent yield surface,

the polycrystal model shows a clear non-normality of the resulting small amount

of plastic flow. Kuroda and Tvergaard (1999) suggested that this apparent non-

normality must be a vertex type of effect resulting from the Taylor model, since

normality is built into the model for each of the slip systems involved. These

theoretical predictions have been verified experimentally for steel and aluminum

alloy sheets, by Kuwabara et al. (2000), and for low carbon steel tube by Ku-

wabara et al. (2003). Since incorporation of vertex-type effects is necessary toobtain good predictions of plastic instability, accurate prediction of the limits to

ductility will require a phenomenological plasticity model that includes an ap-

propriate combination of anisotropic plasticity with vertex-type effects, as pro-

posed recently by Kuroda and Tvergaard (2001).


5. Conclusions

The anisotropic plastic deformation characteristics of a 5154-H112 extruded

aluminum alloy tube have been clarified experimentally for both linear and

combined stress paths, using a servo-controlled tension-internal pressure testing

machine. Successive work contours for different levels of work-hardening aresimilar in shape for linear stress paths (Fig. 6), and the work-hardening behavior

predicted using Hosford’s yield function or the Yld2000-2d yield function under

the assumption of isotropic hardening agrees closely with the observations (Figs. 7

and 8). Moreover, the calculated equivalent stress–equivalent plastic strain curves

based on these yield functions all lie close to a single curve for several combined

stress paths (Figs. 9 and 10). We therefore conclude that the isotropic hardening

assumption is valid for linear and combined stress paths. We also conclude that

both yield functions are effective phenomenological plasticity models for predictingthe anisotropic plastic deformation behavior of the material, although Hosford’s

yield function applies only when the principal stress and anisotropy axes are

parallel.

In this work, the stress states applied to the tubular specimens were confined to

the first quadrant in the principal stress space, where values of both axial and

circumferential stresses are non-negative. In typical tube hydroforming processes

in industry, however, axial compression is often applied to materials during

pressurization, so that the axial stress applied to the material becomes negative.This axial compression is effective in feeding more material into the die cavity and

making the material more easily workable by the pressurization process. To pre-

dict the deformation behavior and the limits to ductility of the material using

numerical simulations, it is therefore also necessary to check the validity of the

material model for deformation modes including axial compression ðr/ < 0Þ and

circumferential elongation (bulging) jointly. However, this deformation mode is

difficult to apply to a tubular specimen using the testing apparatus used in the

present work, because buckling occurs in the axial direction. We are now pre-paring a testing apparatus to apply axial compression and circumferential elon-

gation simultaneously to tubular specimens without causing buckling.

Experimental results and the testing of material models will be presented in a

further paper.

Acknowledgements

We thank Dr. Frederic Barlat of ALCOA Technical Center for informing us of

his latest anisotropic yield function, Yld2000-2d, and Professor Mitsutoshi

Kuroda of Yamagata University for helpful discussions. This work was partly

supported by the Nippon Steel Corporation, the AMADA Foundation for Metal

Working Technology, and MEXT.KAKENHI (15560622), to which we are

grateful.


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Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal...

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