Evolution of Yield Surface in the 2D and 3D Stress Spaces

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Please cite this paper as follows:

Shin-Jang Sung, Li-Wei Liu, Hong-Ki Hong and Han-Chin Wu, Evolution of Yield

Surface in the 2D and 3D Stress Spaces, International Journal of Solids and Structures,

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Evolution of yield surface in the 2D and 3D stress spaces

Shin-Jang Sung

a

, Li-Wei Liu

a

, Hong-Ki Hong

a,⇑

, Han-Chin Wu

a,b

a

Department of Civil Engineering, National Taiwan University, Taipei 10617, Taiwan

b

Department of Civil and Environmental Engineering, University of Iowa, Iowa City, IA, USA

a r t i c l e

i n f o

Article history:

Received 14 September 2010

Received in revised form 13 November 2010 Available online 22 December 2010 Keywords:

Yield surface Yield surface rotation

Automated yield stress determination Axial–torsional-internal pressure experiment

a b s t r a c t

Initial and subsequent yield surfaces for 6061 aluminum, determined by a method of automated yield stress probing, are presented in the 2D (rzzrhz) and 3D (rhhrzzrhz) stress spaces. In the (rzzrhz)

space, yield surfaces at small pre-strains show the noses and unapparent cross effect. At larger pre-strains, they become ellipses with positive cross effect. In the (rhhrzzrhz) space, the initial yield

surface is not well described by von Mises yield criterion due to material anisotropy. The yield surfaces of various torsional pre-strains show obvious rotation around therzzaxis but they do not rotate when

sub-jected to axial pre-strains. Therefore, the rotation behavior of yield surface is pre-strain path dependent. The rotation of yield surfaces in the 3D space is the emphasis of the present paper. Coupled axial–tor-sional behavior subjected to torsion after axial pre-strain are also presented for the same material that is used to determine the yield surfaces. This information is useful for veriﬁcation of constitutive models. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Evolution of yield surface is one of important characteristics of plastic behavior. Three main modes of yield surface evolution, including isotropic expansion or contraction, translation, and dis-tortion, are well recognized by researchers. However, the rotation of yield surface has not been much investigated. The yield surfaces are inﬂuenced by many factors. The most obvious one is the differ-ence in material, including different heat treatments. In addition, the initial anisotropy induced by the manufacturing process is quite important. The anisotropic materials show more complicated behavior under plastic deformation by inducing additional anisot-ropy. For small plastic pstrains of less than 1%, experimental re-sults showed no cross effect (Phillips and Tang, 1972; Phillips et al., 1974; Phillips and Moon, 1977; Moreton et al., 1978; Wu and Yeh, 1991). For plastic pre-strains larger than 1%, two different kinds of evolution of yield surfaces including expanding with positive cross effect and shrinking with negative cross effect were observed (Hecker, 1971; Helling et al., 1986; Khan et al., 2009, 2010a,b; Ng et al., 1979; Shiratori et al., 1973; Stout et al., 1985; Wu, 2003). There were also experimental results which showed contraction ﬁrst and then expansion with increasing plastic pre-strains (Helling et al., 1986; Shiratori et al., 1973; Williams and Svensson, 1971). More information about the evolution of yield surface was reported in Ellis et al. (1983), Boucher et al. (1995), Lissenden and Lei (2004), and in the book (Wu, 2005).

The experimental study of the yield surface was generally con-ducted on plane stress specimens including plate-like specimens or thin-walled tubes. For rolled plates or cross-shaped specimens, tensile tests with different loading axes or biaxial testing were mostly used (Ikegami, 1975a,b; Kreissig and Schindler, 1986; Losilla and Tourabi, 2004). In thin-walled tubes with z denoting the axial direction and h the circumferential direction, the traditional axial–torsional testing was limited to yield loci of the (

r

zz

r

hz)

space with zero hoop stress

r

hh, where

r

zz was the axial stress

and

r

hzwas the shear stress. Additional internal or external

pres-sure was needed to obtain stress states in the (

r

hh

r

zz) space or

in the (

r

hh

r

zz

r

hz) space. Probing subsequent yield surfaces

in the half (

r

hh

r

zz) space, only tension in the hoop stress, can

be accomplished by simply applying axial load and internal pres-sure. These works were presented in Lipkin and Swearengen (1975), Phillips and Das (1985), Khan et al. (2010b). Applying additional external pressure was harder but needed to probe yield surfaces in the whole (

r

hh

r

zz) space, including compression in

hoop stress, and the works were reported by Shiratori et al. (1973) and Moreton et al. (1978). There have been only a few experimental results devoted to the determination of yield surface in the (

r

hh

r

zz

r

hz) space. Shiratori’s group (Shiratori et al.,

1973) investigated subsequent yield surfaces in the (

r

hh

r

zz

r

hz) space by plotting contour lines representing constant

ax-ial stresses or shear stresses in the (

r

zz

r

hz) or (

r

hh

r

zz) space,

respectively. However, data points in that paper were insufﬁcient to draw contour lines in detail.Phillips and Das (1985)determined initial and subsequent yield surfaces in the (

r

hh

r

zz

r

hz) space

and presented yield loci of the yield ellipsoid cut by various planes parallel to the

r

hzaxis.

⇑Corresponding author. Fax: +886 2 27396752. E-mail address:hkhong@ntu.edu.tw(H.-K. Hong).

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A different type of experiment performed by Samanta’s group (Kim, 1992; Kumar et al., 1991; Mallick et al., 1991) on thin-walled tubes was to independently control the axial load, torsion, and internal pressure to simulate uniaxial stressing of a thin-walled element along a direction making an angle

a

with the circumferen-tial direction h of the tube (called off-axis tension byTakeda and Mizukami (2008)). The uniaxial yield stress denoted by

r

u was

determined and plotted against the changing

a

inFig. 1for three conditions of torsion pre-strain. Although not mentioned in their articles, these experimental results actually led to rotation of sub-sequent yield surfaces in the (

r

hh

r

zz

r

hz) space and the details

about the rotation of yield surfaces will be discussed in Section2. In Mallick et al. (1991), the yield surfaces evolved and axes of material orthotropy changed when subjected to torsion pre-strains. But, the yield surfaces were presented in the ﬁrst quadrant of spaces formed by axes of orthotropy and the rotation of yield surface was not apparent in that presentation. Takeda investigated initial and subsequent anisotropy by use of off-axis torsion tests and off-axis tension tests and used the anisotropic yield function to describe the results in various materials (Takeda, 1991, 1993; Takeda and Chen, 2001; Takeda and Mizukami, 2008; Takeda and Nasu, 1991). His articles also revealed that anisotropy might exist in fully annealed (–O) specimens which had been subjected to deformation during manufacturing.

Until now, no systematic experimental results about anisotropic initial and subsequent yield surfaces in the (

r

hh

r

zz

r

hz) space

have been reported in the literature. This information is important in mathematically modeling the plastic behavior of anisotropic materials. In the present study, initial yield surfaces were obtained

in the (

r

hh

r

zz

r

hz) space, which indicated initial anisotropy of

the material tested. After axial pre-strains or torsional pre-strains, subsequent yield surfaces were probed to illustrate the evolution of yield surfaces. Special attention was given to the rotation of yield surfaces. In addition, a set of experiments was conducted to determine the coupled axial–torsional behavior of thin-walled tubes subjected to various boundary conditions. The latter infor-mation is useful for the veriﬁcation of constitutive models. 2. Rotation of subsequent yield surfaces

Mallick et al. (1991) conducted experiments on thin-walled tubes with axial force, torsion, and internal pressure independently controlled to simulate pure tension of a thin-walled element along a direction making an angle

a

with the circumferential direction h of the tube. The axial direction of the tube is denoted by z. The uni-axial tensile stress increased monotonically and the yield stress

r

u

was determined for each value of

a

. Thus,

r

uvs.

a

curves were

plot-ted and shown in Fig. 1 for three kinds of specimens: (1) as-received; (2) specimens subjected to torsional pre-strains of

c

= 0.1; and (3) specimens subjected to torsional pre-strains of

c

= 0.3.

The yield stress

r

u can be transformed into a set of stress

components (

r

hh,

r

zz,

r

hz) by use of

r

hh¼

r

ucos2

a

;

r

zz¼

r

usin2

a

;

r

hz¼

r

usin

a

cos

a

: ð1Þ

Using these stress components a point can be plotted in the (

r

hh

r

zz

r

hz) space, with

r

hhand

r

zzforming the biaxial plane,

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and

r

hzthe vertical axis normal to it. The initial yield surface for an

isotropic material is an ellipsoid in this space. But, the anisotropic yield surfaces are distorted ellipsoids. The set of points obtained by (1) are points on the surfaces of distorted yield ellipsoids and the positions of these points can be best viewed by the two projec-tions. The ﬁrst projection is onto the (

r

hh

r

zz) plane (the plan

view) and the second projection (the elevation view) is onto a plane that passes through the

r

hz axis and the S2-direction which is

bisecting the right angle formed by the

r

hhand

r

zzaxes. The S2

-direction is the -direction of the minor principal axis of the yield ellipsoid of isotropic material. The elevation view would show clearly if there is a rotation of the yield ellipsoid around its major principal axis. After transforming the experimental

r

uvs.

a

curves

inMallick et al. (1991)into the (

r

hh

r

zz

r

hz) space by Eq.(1),

plan view and elevation view are plotted as shown inFigs. 2 and 3. To our surprise, rotation of the yield surface was observed in the elevation view. Results showed that the yield surfaces of aniso-tropic material rotated even with a proportional pre-strain path in torsion. The rotation of yield surface had been discussed in some articles but it was only in the (

r

zz

r

hz) or (

r

hh

r

zz) spaces with

non-proportional paths (Ishikawa, 1997;Kaneko et al., 1976; Shiratori et al., 1974). Although a slight rotation was found in Fig. 12 ofPhillips and Das (1985), the rotation of the yield surface was not discussed in that paper. Because of insufﬁcient experimen-tal results in the (

r

hh

r

zz

r

hz) space, it gave us the motivation to

ﬁnd out the rotation of subsequent yield surfaces in the (

r

hh

r

zz

r

hz) space.

3. Specimens and test equipment

The specimens used in experiments were 6061 aluminum alloy tubes, designed for combined loading tests subject to axial, torsion and internal pressure. The gauge section of the specimen was 60 mm long with an outer diameter of 25 mm. The inner diameter was 22 mm and the wall thickness was 1.5 mm with a ratio of ra-dius to thickness of 8.33.Fig. 4shows the dimensions of the spec-imen. After machining, the specimens were annealed at a temperature of 410 °C for 2.5 h and the temperature was then low-ered slowly to 275 °C in 5 h, and ﬁnally they were furnace-cooled to room temperature. In addition, two steel plugs were installed at the ends of tubes and one of them was designed to allow ma-chine oil to run through, in order to apply internal pressure.

The testing machine used for the experiments was a MTS 809 servo hydraulic axial–torsional material testing system. A self-made hydraulic system was used to apply internal pressure, which was an open-loop control system. The load capacity was ±500 kN in tension, ±5500 N-m in torsion and 35 MPa in internal pressure. The axial stress

r

zz was obtained by dividing the axial force by

the cross-sectional area of the thin-walled specimen. The shear stress

r

hzdue to torsion was determined at the mid-thickness of

the thin-wall. An MTS 632.80c-04 axial–torsional extensometer with a gauge length of 25 mm was mounted on the specimen to measure axial and shear strains. A PC with LabVIEW 8.6.1 was used to give command voltage to the MTS console and collected signals by NI PCI-6289 DAQ, a data acquisition card which had 4 analog output channels and 32 analog input channels. The card met the demand of controlling 3 channels and acquiring data from 8 sen-sors simultaneously. The real-time calculation about automated yield point determination was also executed on it.

4. An automated yield stress determination

In earlier experimental determination of yield surface, the pro-cess was not fully automated. Investigators had to use their judg-ments in the determination of yield points. A fully automated method for the yield point determination was introduced and ap-plied to the research work reported in this paper.

Because of special characteristics of servo-controlled hydraulic systems, the loading method could not be the same as that used for dead-load machines. The constant rate loading method was adopted when determining yield points. However, the traditional criterion of yielding, deﬁned by deviation from elastic linearity, needed modiﬁcation because of data scatter in the servo-controlled system.

The scatter of data was affected by factors related to the equip-ment such as actuators, the extensometer, the hydraulic system, etc. It was common that the scatter of data varied during an exper-iment. In the constant rate loading method, if the difference of scatter was too large in one experiment, the results of experiments with traditional yield criterion would be poor. Therefore, for the purpose of automation, scatter of data must be taken into consid-eration by way of probability when determining yield points.

For each experiment, a pre-test of loading/unloading was con-ducted within the range of one-third yield stress to eliminate any contact gaps between the specimen and the test equipment. Then,

Fig. 2. The plan view of three torsional pre-strains, transformed from experimental data ofFig. 1.

Fig. 3. The elevation view of three torsional pre-strains, transformed from experimental data ofFig. 1.

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in each test, data of the ﬁrst 4–6 MPa were collected to evaluate model parameters, such as slope and intercept, by linear regres-sion. After calculating the parameters, deviations ddevof those data

from the ﬁtted straight line were calculated from

ddev¼

r

ðobservedÞij

r

ðfittedÞ ij

r

ðobservedÞ ij

r

ðfittedÞ ij h i1=2 ð2Þ

Because the controlled paths in the reported experiments were mainly related to the axial and shear strains, and not to the hoop strain, Eq.(2)was expanded as

ddev¼

r

zz

r

hz ðobservedÞ E 0 0 2G

_e

zz

e

hz ðobservedÞ þ bzz bhz ! ð3Þ

where

e

zz,

e

hzwere axial strain and shear strain, respectively; E and

G were Young’s modulus and shear modulus, respectively; and bzz,

bhzwere the components of ‘‘zero offset stress’’.

The controlled paths were designed to move either along the axial or torsional direction so that only one of E and G in Eq.(3) was used when ﬁtting the experimental curve to the controlled path. The unused E or G would be set to zero to prevent unneces-sary noise. The scatter of ddevcalculated from each point of the ﬁrst

4–6 MPa was studied by Weibull distribution (seeAppendix A). Two independent normal distributions with close standard devia-tions were obtained for axial and torsional direcdevia-tions, respectively. The 95% conﬁdence intervals were used to determine the two parameters of Weibull distribution. Within this model, 99% cumu-lative probability was taken to evaluate the quantiﬁed scatter of data, called dest(seeFig. 5).

For a speciﬁed offset strain

e

offset, the yield point was A, with an

offset stress of dobj(seeFig. 5). However, due to data scatter, the

point when ddev= dobjwas at B, which had to be corrected by dest

to obtain A. Note that dðaxialÞ

obj ¼ E

e

offset; dðtorsionÞobj ¼ G

c

offsetand dest

was the scatter of data as previously explained. Thus, the expres-sion either

dðaxialÞ_offset ¼ dðaxialÞobj d ðaxialÞ est ð4Þ or dðtorsionÞoffset ¼ d ðtorsionÞ obj d ðtorsionÞ est ð5Þ

was used to determine yield point A. In the experiments,

e

off-set= 12.5

l

,

c

offset= 25

l

, E = 72 65 GPa and G = 25 23 GPa. E

and G varied with pre-strain. If an equivalent offset strain was de-ﬁned by

e

¼

e

ij

e

ij 1=2 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

e

2 11þ 2

e

212 q ð6Þ

then, in the axial probing

e

¼

e

offset, and in torsion probing

e

¼

c

offset=

ffiffiffi 2 p

. By use of

c

offset= 25

l

, the equivalent strain was

found to be

e

¼ 17:68

l

. Thus, in the experiments

e

–

e

offsetand

to-gether with the variation of E led to a variation of, dðaxialÞ obj in(4).

Similarly, there was a variation of dðtorsionÞ

obj in(5), because of variation

of G. Therefore, Eqs.(4) and (5)become

dðaxialÞ_offset ¼ ð0:813 1:273Þ dðaxialÞest ðMPaÞ ð7Þ

dðtorsionÞ_offset ¼ ð0:575 0:625Þ dðtorsionÞest ðMPaÞ ð8Þ

In(7), 0.813 was found from E = 65 GPa and

e

offset= 12.5

l

; 1.273

was found from E = 72 GPa and

e

offset= 17.68

l

. These were the

ex-treme cases. In(8), 0.575 was found from G = 23 GPa and 0.625 was found from G = 25 MPa, both with

c

offset= 25

l

. The variation of

0.46 MPa in dðaxialÞ_offset and 0.05 MPa in dðtorsionÞ_offset did not lead to noticeable differences in the yield surfaces determined. The yield point was found when ddev= doffset. To make the determination more robust,

not only one data point but a certain number of sequential data points were used. The loading rate would also decrease to one-third of original one when ddevP0.6doffset. It meant that the loading was

slower when approaching the yield point. In this way, the penetra-tion of probing into the plastic region could be minimized.

In all experiments in this research, destwas between 0.25 MPa

and 0.45 MPa and the scatter of internal pressure was within ±0.06 MPa, which could be converted to a ±0.5 MPa hoop stress. In probing yield surfaces, the strain rates were about _

e

zz¼ 2 106s1 and _

c

¼ 2_

e

zh¼ 5:5 106s1. Note that it was

mentioned byEllis et al. (1983) and Wu and Yeh (1991)that the probing rate had little effect on the yield surface determination. This automated process of yield stress determination required less manipulation by the operator and it had higher efﬁciency and de-creased artiﬁcial errors.

5. Experimental program

This research was concentrated in the investigation of evolution of yield surfaces in the (

r

zz

r

hz) space and in the (

r

hh

r

zz

r

hz)

space. However, additional experiments were also conducted on the same material to investigate coupled stress–strain behavior in the combined tension–torsion tests of tubular specimens. This

Fig. 4. Test specimens.

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information is useful for further study of plastic deformation and responses.

5.1. Yield surfaces in the (

r

zz

r

hz) space

Without internal pressure, the initial yield surfaces and the sub-sequent ones subjected to various axial or torsional pre-strains were probed. The probing path like a ﬁshbone was O-1-2-3-4-5-6-7-8-9-10-11-12-3-4-O and changed its direction at O, A, B, C, D (Fig. 6). The double probing in the pre-strained direction was be-cause of unstable yield stresses on the reverse loading side.Fig. 6 shows the probing paths for specimens with axial pre-strains (path A) and specimens with torsional pre-strains (path B). When prob-ing yield surfaces, the control channels were axial strain and rota-tion. When applying torsional pre-strains, however, the axial direction was load controlled to keep zero axial stress. This condi-tion is called free-end torsion. Remounting the extensometer was needed because its range of shear strain measurement was

c

= 2.18%. Because of zero axial stress (free-end), specimens were elongated when applying torsional pre-strains. Therefore, the total shear strain was

c

all¼

c

1=ð1 þ

e

1Þ þ

c

2=ð1 þ

e

2Þ þ þ

c

n=ð1 þ

e

nÞ ð9Þ

where

c

allwas the shear strain after remounting the extensometer n

times.

e

iand

c

iwere strain readings from the extensometer at the

end of remounting number i. However, the change of the axial strain was much smaller than 1 for this experiment subjected to torsional pre-straining so that

c

allcould just be the sum of each shear strain.

After pre-straining, the stress was reduced to a point close to the center of the subsequent yield surface, and then the specimens were relaxed for one hour before probing the yield surface.

5.2. Yield surfaces in the (

r

hh

r

zz

r

hz) space

In the (

r

hh

r

zz

r

hz) space, the load control was applied in the

axial direction with increasing or decreasing internal pressure. In that manner, if the internal pressure p was speciﬁed, the axial stress and hoop stress of the specimen were

r

zz¼

p

r2_p

p

ðR2 r2_Þ¼ p ðR=r þ 1ÞðR=r 1Þ ð10Þ

r

hh¼ 2pr 2ðR rÞ¼ p R=r 1 ð11Þ

where R denotes the outer radius and r the inner radius, respec-tively. The ratio of

r

hhto

r

zzwas (R/r + 1), which was close to 2

un-der thin-walled tube assumption. One special caution was that

r

zz

should not be read directly from the load cell mounted on the

MTS frame when applying internal pressure. Because of load con-trol, the value read out from the load cell was constant during changing internal pressure. However, the axial stress actually had varied due to the change of internal pressure. Therefore, after the internal pressure had stabilized, the reference point (zero point) of the load cell was altered to the value that the specimen was actu-ally subjected to. When pressurizing internactu-ally, the loading moved along the path of

r

hh= 2

r

zz, shown by dotted line in the

(

r

hh

r

zz

r

hz) space ofFig. 7. The shear stress

r

hzwas again

deter-mined at the mid-thickness of the thin-wall.

With or without torsional pre-strain and at a speciﬁed hoop stress, the probing path in the (

r

hh

r

zz

r

hz) space was

A-1-2-3-4-3-4-5-6-5-6-7-8-7-8-A, shown in Fig. 7. In the experiment, the internal pressure was ﬁrst increased and the stress point moved from center of yield surface O to point A inFig. 7. The stress point moved from A to B, C or D by applying tensile or compressive axial stress; and it moved along the shear stress direction such as B3, B4, and C5 by applying torsion. The probing path lay in a plane parallel to the (

r

zz

r

hz) plane. The probing paths at different

con-stant hoop stresses lay in different parallel planes. The yield points along the torsional direction were probed twice to obtain more data points. A, B, C, D were special points with meanings of

r

hh= 2

r

zz,

r

hh=

r

zz,

r

hh=

r

zz,

r

zz= 0, respectively. The union of B

points was a straight line called the S1axis, which bisects the right

angle formed by the

r

hhand

r

zzaxes. The union of C points was

an-other line called the S2axis (same as S2-direction in Section 2),

which bisects the right angle formed by the

r

hhand

r

zzaxes. After

obtaining yield surfaces in sections with different hoop stresses and parallel to the (

r

zz

r

hz) space, experimental data could be

shown in four special stress spaces, called (

r

hh

r

zz), (S1

r

hz),

(

r

hh

r

hz), (S2

r

hz), to demonstrate the ellipsoidal yield surface

in the (

r

hh

r

zz

r

hz) space. By considering the results of this

sec-tion and Secsec-tion5.1together, ellipsoidal yield surfaces of different torsional and axial pre-strains in the (

r

hh

r

zz

r

hz) space could

be clearly visualized.

The S1and S2-directions are respectively the major and minor

principal directions of the yield ellipsoid of the isotropic material. In the present case of anisotropic material, these were not principal directions, but these directions were helpful in the visualization of the evolved 3D yield ellipsoid. The yield curves in the aforemen-tioned (

r

hh

r

zz), (S1

r

hz), (

r

hh

r

hz), (S2

r

hz) stress

sub-spaces were the intersection curves of the yield ellipsoid was cut by the four planes. In addition, the yield curves presented in Sec-tion5.1were the yield ellipsoid was cut by the (

r

zz

r

hz) plane.

5.3. Tension–torsion tests

In addition to evolution of the yield surface, tension–torsion tests were conducted by displacement and rotation control in each

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control channel to study plastic behavior at large deformation. The specimens were subjected to various tensile strains ﬁrst. Then, they were subjected to torsion with increasing shear strain while keeping the length of specimen constant (ﬁxed-end torsion). The test continued until buckling occurred. Remounting the extensom-eter was needed in this experiment because the measuring range of strains for the extensometer were only

e

= 4.8% and

c

= 2.18%. During axial pre-straining, the measured axial strain should be modiﬁed due to remounting, as:

e

all¼ ð1 þ

e

1Þð1 þ

e

2Þ ð1 þ

e

nÞ 1 ð12Þ

where

e

allwas the axial strain after remounting and

e

iwere axial

strain readings from the extensometer at the end of remounting number i. During ﬁxed-end torsion, the measured shear strain was the sum of all shear strain readings of all re-mountings. Eq. (9)did not apply in this case.

The results of these tests can be used to verify constitutive mod-els. Although similar tests have been conducted by other investiga-tors, the present tests were conducted on the same material as that used in the determination of the yield surfaces in Sections5.1 and 5.2. Therefore, material parameters of the constitutive models can be determined.

6. Results and discussions 6.1. The (

r

zz

r

hz) space

The initial yield surfaces were probed on most specimens before pre-straining. Results of 10 specimens are shown inFig. 8. The dot-ted circle represents the von Mises yield surface where the axial and shear yield stresses are ±34 MPa and ±19.6 MPa, respectively. Only one specimen was used to obtain subsequent yield sur-faces with axial pre-strains of 0.5%, 1.0%, 3.0% and 4.0%. Experimen-tal results of axial pre-strains 0.5% and 1.0% are shown inFig. 9 together with the initial yield surface. Results of axial pre-strains 3.0% and 4.0% are shown inFig. 10. All yield surfaces of different pre-strains are summarized inFig. 11. The yield surfaces of axial pre-strain 0.5% are similar to flattened ellipses with flattened rear parts and rounded nose. This characteristic is not clearly shown for the cases with the axial pre-strains of 3% and 4%. The rear sides of axial pre-strains 3% and 4% almost coincide. However, it is interest-ing that the size of yield surfaces decreased first and then increased after an axial pre-strain of 1.0%.

After probing the subsequent yield surface with an axial pre-strain of 4%, the specimen was unloaded to zero axial stress. Then, it was loaded again to reach the same axial stress as previously ob-tained with an axial pre-strain of 4%. The subsequent yield surface was probed again and the result is shown together with the previ-ous one inFig. 12. The yield surface determined after reloading was a little larger but it could be regarded as a good approximation of the original yield surface with an axial pre-strain of 4%. This exper-imental evidence showed that, despite possible inaccuracy men-tioned by Khan et al. (2009), the yield surface probed after unloading followed by reloading to its original stress would pro-vide a good approximated yield surface (Wu, 2003).

Six specimens were used to determine subsequent yield sur-faces with torsional pre-strains of 0.144%, 0.5%, 1.0%, 3.0% and 6.0%. Experimental results of axial pre-strains 0.144% and 0.5% are shown inFig. 13. In each case of 1.0%, 3.0% and 6.0% pre-strains, experiments were conducted on two specimens. Results of tor-sional pre-strains 1.0%, 3.0% and 6.0% are shown in Figs. 14–16, respectively. All yield surfaces with different pre-strains are sum-marized inFig. 17. The behavior of this set of yield surfaces was

Fig. 8. The initial yield surface.

Fig. 9. The subsequent yield surfaces of axial pre-strains 0.5% and 1.0% compared with the initial yield surface.

Fig. 10. The subsequent yield surfaces of axial pre-strains 3.0% and 4.0% compared with the initial yield surface.

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Fig. 11. A summary of yield surfaces fromFigs. 8 and 9.

Fig. 12. Comparison of yield surface probed after reloading to that probed at initial loading.

Fig. 13. The subsequent yield surfaces of torsional pre-strains 0.144% and 0.5% compared with the initial yield surface.

Fig. 14. The subsequent yield surface of torsional pre-strain 1.0% compared with the initial yield surface.

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similar to those with axial pre-strains. The shapes of yield surfaces gradually became ellipses and 1.0% torsional pre-strain was also a dividing point about the size of the yield surface and cross effect. The size of the yield surface decreased ﬁrst at small pre-strain and increased when the pre-strain was larger than 1.0%. The subse-quent yield surface of a reversed torsional pre-strain 3% was also probed and the results compared with the one with a pre-strain of 3% in Fig. 18. It is seen that the results of the two cases are symmetric with respect to the

r

zzaxis.

6.2. The (

r

hh

r

zz

r

hz) space

The initial yield surfaces probed on two specimens in the (

r

hh

r

zz

r

hz) space are presented in the (

r

hh

r

zz), (S1

r

hz),

(

r

hh

r

hz) and (S2

r

hz) spaces, and shown inFigs. 19–22,

respec-tively. Results of the two specimens were quite compatible. Initial anisotropy was observed in the (

r

hh

r

zz) space in which the hoop

Fig. 17. A summary of yield surfaces fromFigs. 12–15.

Fig. 18. The subsequent yield surfaces of torsional pre-strain 3.0% and 3.0% compared with the initial yield surface.

Fig. 19. The initial yield surface in the (rhhrzz) space.

Fig. 20. The initial yield surface in the (S1rhz) space.

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Fig. 22. The initial yield surface in the (S2rhz) space.

Fig. 23. The subsequent yield surface of torsional pre-strain 0.144% in the (S1rhz)

space.

Fig. 24. The subsequent yield surface of torsional pre-strain 0.144% in the (rhhrhz) space.

space.

Fig. 27. The subsequent yield surface of torsional pre-strain 1.0% in the (rhhrhz)

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space.

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yield stress was larger than the axial yield stress. The von Mises yield criterion which described the initial yield surface well in the (

r

zz

r

hz) space was not able to describe the initial yield

sur-face in the (

r

hh

r

zz

r

hz) space.

With different torsional pre-strains of 0.144%, 1.0%, 3.0% and 6.0%, yield surfaces in the (S1

r

hz), (

r

hh

r

hz) and (S2

r

hz)

spaces were experimentally obtained. In these cases, except for 0.144% which had only one specimen, two specimens were used to determine the experimental results of each case. All data points are shown in ﬁgures,Figs. 23–25for 0.144% pre-strain;Figs. 26–28 for 1.0% pre-strain;Figs. 29–31for 3.0% pre-strain andFigs. 32–34 for 6.0% pre-strain. The results in the (S1

r

hz), (

r

hh

r

hz) and

(S2

r

hz) spaces are summarized inFigs. 35–37, respectively. The

clockwise rotation of yield surfaces was observed clearly in these ﬁgures. The amounts of yield surface rotations were almost the same in the (S1

r

hz), (

r

hh

r

hz) and (S2

r

hz) spaces although

the rotation was slightly larger in the (

r

hh

r

hz) space. Therefore,

it is fair to say that the yield surface rotated around the

r

zzaxis.

The positive cross effect is more obvious than that in the (

r

zz

r

hz)

space when torsional pre-strain was 1.0%.

The test of torsional pre-strain 3% was also conducted to ob-serve the behavior of yield surface in the (

r

hh

r

zz

r

hz) space

un-der a reverse torsion. Yield surfaces with torsional pre-strains of 3% and 3% are shown together inFigs. 38–40. It is seen that the rotation for specimen with 3% pre-strain was clockwise and that for a specimen with 3% pre-strain was counterclockwise.

Yield surfaces of axial pre-strains in the (

r

hh

r

zz

r

hz) space

were obtained in one specimen. The specimen was subjected to 4% axial pre-strain ﬁrst and then unloaded to a zero axial stress. After ﬁnding out the approximated center of the subsequent yield surface, which was the axial stress of 12.5 MPa in this test, the cen-ter was set as the starting point of S1and S2axes. The yield surfaces

probed in the (S1

r

hz) and (S2

r

hz) spaces did not rotate, and are

shown together with the initial yield surfaces inFigs. 41 and 42, respectively. These experimental results showed that the rotation of subsequent yield surfaces was pre-strain path dependent. 6.3. Tension–torsion tests

In the case of free-end torsion, one specimen without axial pre-strain was subjected to a shear pre-strain of 6%. Results in the (

c

r

hz)

space.

Fig. 35. A summary of yield surfaces in the (S1rhz) space fromFigs. 19, 22, 25, 28

and 31.

Fig. 36. A summary of yield surfaces in the (rhhrhz) space fromFigs. 20, 23, 26, 29

and 32.

Fig. 37. A summary of yield surfaces in the (S2rhz) space fromFigs. 21, 24, 27, 30

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and (

c

e

zz) spaces are shown inFigs. 43 and 44. It is seen that the

specimen was elongated in the case of free-end torsion.

In the case of ﬁxed-end torsion, ﬁve specimens were subjected to different axial pre-strains. Three of these specimens were from a different batch of specimens (Batch B of the same material), not from the same batch used to probe yield surfaces (Batch A). Spec-imens of Batch B were annealed at 415 °C in 3 h and then furnace-cooled to room temperature. With axial pre-strains of 0.74% (Batch A), 2.5% (Batch B), 4.0% (Batch B), 10% (Batch A) and 10% (Batch B) at _

e

zz¼ 3 8 105s1, the curves in the (

e

zz

r

zz) space are

shown inFig. 45. The curve of 10% (Batch B) was a little different from others but results of the two batches were compatible. With each different axial pre-strain, specimens were subjected to a shear strain of up to 13% at _

c

¼ 1 2 105s1_{. Results in the (}

_c

_r

hz)

and (

c

r

zz) spaces are shown inFigs. 46–48. The effect of axial

pre-strains was obvious in the subsequent torsion. In the (

c

r

hz)

space, curves of smaller axial pre-strains are lower. In the (

c

r

hz)

space, axial stresses decreased rapidly after torsion started and they approached stable values with increasing shear strains. If buckling did not happen, axial stresses would go into small com-pression at large shear strains, as in the cases of 0.74% (Batch A)

Fig. 38. The subsequent yield surfaces of torsional pre-strain 3.0% and 3.0% in the (S1rhz) space.

Fig. 39. The subsequent yield surfaces of torsional pre-strain 3.0% and 3.0% in the (rhhrhz) space.

Fig. 40. The subsequent yield surfaces of torsional pre-strain 3.0% and 3.0% in the (S2rhz) space.

Fig. 41. The subsequent yield surface of axial pre-strain 4.0% in the (S1rhz) space.

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Fig. 43. The shear stress–strain curve of free-end torsion without axial pre-strain.

Fig. 44. Theezzvs.ccurve of free-end torsion without axial pre-strain.

Fig. 45. Axial stress–strain curves of various specimens.

Fig. 46. Shear stress–strain curves for ﬁxed-end torsion.

Fig. 47. Axial stress distribution for ﬁxed-end torsion.

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and 2.5% (Batch B). Buckling occurred at large shear strains as in the cases of 4.0% (Batch B) and 10% (Batch A, B) pre-strains. The dif-ferences between the two batches of specimens can be observed clearly from the curves of 10% pre-strain inFigs. 47 and 48. 7. Conclusions

Three series of experiments were conducted using annealed 6061 aluminum alloy tubes with a servo hydraulic testing system. They were (1) evolution of yield surfaces in the (

r

zz

r

hz) space;

(2) evolution of yield surfaces in the (

r

hh

r

zz

r

hz) space; (3)

combined tension–torsion tests. An automated yield stress deter-mination was used when probing yield surfaces to increase efﬁ-ciency and decrease artiﬁcial errors. The following phenomena have been observed from experimental results.

1. Yield surfaces of various axial or torsional pre-strains in the (

r

zz

r

hz) space showed noses and unapparent cross effect at

small pre-strains. They became ellipses and had positive cross effect with increasing pre-strains. In addition, the size of yield surfaces decreased ﬁrst and then increased after an axial pre-strain of 1.0%.

2. Initial anisotropy was observed from the initial yield surface in the (

r

hh

r

zz

r

hz) space. The von Mises yield criterion that

seems suitable in the (

r

zz

r

hz) space is actually inadequate

for the (

r

hh

r

zz

r

hz) space.

3. The rotation of the yield surface was pre-strain path dependent. The clockwise rotation of subsequent yield surfaces shown in the (S1

r

hz), (

r

hh

r

hz) and (S2

r

hz) spaces was observed

with various torsional pre-strains. If subjected to a reverse tor-sion, yield surfaces in these stress spaces would rotate in a reverse direction. On the other hand, experiments showed that, if subjected to axial pre-strains, yield surfaces did not rotate around the

r

zz axis. Therefore, the rotation behavior of yield

surface is pre-strain path dependent and a theory of plasticity should include a way to account for rotation of the yield surface. 4. A free-end torsion of a thin-walled cylindrical specimen gave rise to specimen elongation. In the combined tension–torsion, with ﬁxed-end condition, the shear stress increased while the axial stress decreased rapidly at small strain levels with increas-ing shear strain.

Acknowledgement

This research is supported by National Research Council of Tai-wan (NSC 97-2221-E-002-120-MY2, 2811-E-002-118 and 98-2811-E-002-090).

Appendix A

Taking one set of data during axial loading as an example, the scatters of signals are shown as normal distributions in either axial or shear stress, seeFigs. A.1 and A.2. Their probability density func-tions are faxial¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

p

d2axial q exp

r

ðobservedÞ zz

r

ðfittedÞzz 2 2d2axial 0 B @ 1 C A ðA:1Þ ftorsion¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

p

d2torsion q exp

r

ðobservedÞ hz

r

ðfittedÞ hz 2 2d2torsion 0 B @ 1 C A ðA:2Þ

where d is standard deviation. The subscripts ‘‘axial’’ and ‘‘torsion’’ denote, respectively, the axial direction and torsion direction. If two

normal distributions are independent and daxial= dtorsion, the

proba-bility density function of occurrence of ddevis

g ¼ 2

p

ddevfaxialftorsion¼

ddev d2 exp d2dev 2d2 ! ðA:3Þ

where ddev¼

r

ðobservedÞzz

r

ðfittedÞzz

2 þ

r

ðobservedÞ hz

r

ðfittedÞ hz 2 1=2 and d = daxial= dtorsion. By setting k = 2, x = ddev, and k ¼

ffiffiffi 2 p

d, the proba-bility density function of Weibull distribution

W ¼k k x k k1 exp x k kk ; x P 0 ðA:4Þ

becomes Eq.(A.3).

Fig. A.1. The probability plot of rðobservedÞ zz rðfittedÞzz

and ﬁtted normal distribution.

Fig. A.2. The probability plot of rðobservedÞ hz r

ðfittedÞ hz

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The actual ﬁtted parameters used in this probing were daxial= 0.1005(MPa) and torsion = 0.091(MPa). These values of each

probing are different since the scatter of data vary during the experiment. The values of two standard deviations were close; therefore the ﬁtted result of ddevas a Weibull distribution was good

with parameters k = 1.9581 and k = 0.1349(MPa), shown inFig. A.3. After ﬁnding out the parameters, dest= 0.292(MPa) was chosen

(99% cumulative probability). By plotting the data points for curve ﬁtting directly, shown inFig. A.4, it was seen that the range of var-iation was not large and did not increase rapidly. If plotting sequential points after the curve ﬁtting, shown inFig. A.5, an obvi-ous increase appeared between point 6000 and point 7000. After the yield point had been decided within the range, the loading was stopped for a while before unloading so ddevdid not increase

after point 7000.

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