multiaxial tube expansion test method for measurement of sheet metal deformation behavior under...

International Journal of Plasticity 45 (2013) 103–118

Contents lists available at SciVerse ScienceDirect

International Journal of Plasticity

journal homepage: www.elsevier .com/locate / i jp las

Multiaxial tube expansion test method for measurement ofsheet metal deformation behavior under biaxial tension fora large strain range

0749-6419/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijplas.2012.12.003

⇑ Corresponding author. Tel.: +81 42 388 7083; fax: +81 42 385 7204.E-mail addresses: [email protected] (T. Kuwabara), [email protected] (F. Sugawara).

Toshihiko Kuwabara a,⇑, Fuminori Sugawara b

a Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho,Koganei-shi, Tokyo, Japanb Department of Mechanical Systems Engineering, Graduate School of Engineering, Tokyo University of Agriculture and Technology, 2-24-16 Nakacho, Koganei-shi,Tokyo, Japan

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 July 2012Received in final revised form 26 October2012Available online 27 December 2012

Keywords:A: FractureA: Yield conditionB: Constitutive behaviorC: Mechanical testingForming limit analysis

A servo-controlled multiaxial tube expansion testing machine was developed to measure themultiaxial plastic deformation behavior of sheet metals for the range of strain from initial yieldto fracture. The testing machine is capable of applying arbitrary principal stress or strain pathsto a tubular specimen using an electrical, closed-loop servo-control system for axial force andinternal pressure, in addition to having a novel strain measurement apparatus for tubular spec-imens. Tubular specimens with an inner diameter of 44.6 mm were fabricated from cold rolledinterstitial-free steel sheet with a thickness of 0.7 mm by roller bending and laser welding.Many linear stress paths in the first quadrant of stress space were applied to the tubular spec-imens to measure the forming limit curve (FLC) and forming limit stress curve (FLSC) of the as-received sheet material, in addition to contours of plastic work and directions of plastic strainrates. Results calculated using the Yld2000-2d yield function with an exponent of 6 providedthe closest agreement with the measured work contours and directions of plastic strain ratesfor an equivalent plastic strain range of 0.005–0.36. Moreover, a Marciniak–Kuczynski-typeforming limit analysis was performed using the most appropriate yield function with theassumption of isotropic hardening; the calculated and measured FLC and FLSC were in fairagreement. Therefore, the multiaxial tube expansion test is effective to measure the multiax-ial deformation behavior of sheet metals for a large range of plastic strain.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The establishment of trial-and-error-less manufacturing enhanced by forming simulations, such as finite element analysis(FEA), has been strongly desired in industry to shorten the product development period and reduce costs for prototype man-ufacturing. Improvement of the predictive accuracy for defect formation, such as fracture and springback, using FEA, is a keyto realize trial-and-error-less manufacturing. Thus, it is essential to establish a highly accurate material test method that iscapable of validating the constitutive equations used in the FEA. In sheet metal forming processes, metal sheets are subjectedto various multiaxial stress states. Therefore, the validity of the constitutive equations used in the FEA should also bechecked by multiaxial stress tests (Kuwabara, 2007; Banabic et al., 2010).

In the field of experimental plasticity, the study of material behavior under biaxial loading has been a topic of interest fordecades. Most previous experimental investigations of the yield locus shapes of metals have been conducted using

http://dx.doi.org/10.1016/j.ijplas.2012.12.003

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.ijplas.2012.12.003

http://www.sciencedirect.com/science/journal/07496419

http://www.elsevier.com/locate/ijplas

104 T. Kuwabara, F. Sugawara / International Journal of Plasticity 45 (2013) 103–118

thin-walled tubes with tension–torsion and/or tension–internal pressure loading (Hecker, 1976; Michno and Findley, 1976;Ikegami, 1979; Stout et al., 1983; Khan et al., 2007b; Khan and Liu, 2012). Recently, servo-controlled hydraulic tube bulgetesting machines have been developed, and the path-dependence of forming limit stresses was experimentally investigatedfor extruded aluminum alloy tubes (Kuwabara et al., 2005; Yoshida et al., 2005; Korkolis and Kyriakides, 2008a, 2008b, 2009)and a steel tube (Yoshida and Kuwabara, 2007). Kuwabara et al. (2003) applied the abrupt strain path change method pro-posed by Kuroda and Tvergaard (1999) to a steel tube to successfully measure a subsequent yield locus following equibiaxialtension without unloading and a yield vertex at the point of loading.

The most popular biaxial material test methods for sheet metals are the hydraulic bulge test (Bell et al., 1967; Bramleyand Mellor, 1966; Mellor and Parmar, 1978) and plane-strain tension test (Wagoner and Wang, 1979; Wagoner and Lauk-onis, 1983; Kuwabara and Ikeda, 2002; An et al., 2011). The hydraulic bulge test is useful to measure the stress–strain curvesof sheet metals for a large strain range. However, the stress state is fixed to approximately equibiaxial; therefore, it cannot beused to confirm the validity of constitutive laws for various stress or strain paths. The stress ratio for the plane-strain tensiontest is limited to approximately 2:1 (stress in the tensile direction vs. stress in the width direction); however, the stress com-ponent in the width direction cannot be determined. Moreover, it is difficult to obtain uniform plane-strain deformation inthe gauge area; therefore, the accuracy of the stress measurement is questionable, in particular for a large strain range. Biax-ial compression tests of stacked sheet metals have also been popular (Lee and Backofen, 1966; Kelley and Hosford, 1968;Tozawa, 1978; Khan and Wang, 1990; Khan et al., 2007a; An et al., 2011; Khan and Liu, 2012). However, such tests sufferfrom the (undetermined) effect of friction; therefore, measurement of an accurate stress–strain relationship is difficult. Fur-thermore, the test results may be questionable when the yield of the material may be affected by hydrostatic pressure (Low-den and Hutchinson, 1975; Spitzig and Richmond, 1984), and/or the test material exhibits tension-compression asymmetry(Lou et al., 2007; Kuwabara et al., 2009; Dunand et al., 2012).

Simple shear test (Miyauchi, 1984) and combined simple shear-tension test (Vegter and van den Boogaard, 2006; Mohrand Oswald, 2008; Mohr et al., 2010; An et al. 2011; Dunand et al., 2012) is useful to determine a 3-dimensional yield surfacein the stress space (rx,ry,sxy), where rx and ry are the normal stress components in the rolling and transverse directions,respectively, and sxy is the corresponding shear stress component. However, this test method alone is not sufficient for aquantitative evaluation of the most suitable yield function for a given material; several linear stress path tests in the stressspace (rx,ry,0) are necessary to determine the optimum exponent for a higher order yield function, e.g. the Yld2000-2d yieldfunction (Barlat et al., 2003; Yoon et al., 2004), that has the closest agreement with a measured work contour, as demon-strated by Kuwabara et al. (2011) and Yanaga et al. (2012).

Kuwabara and coworkers developed a biaxial tensile test method for sheet metals using a cruciform specimen (Kuwabaraet al., 1998, 2000) and successfully determined appropriate anisotropic yield functions for industrial use sheet metals (Kuwa-bara et al., 2000, 2002; Kuwabara and Nakajima, 2011; Verma et al., 2011; Andar et al., 2012). Moreover, it has been demon-strated that biaxial tensile testing with a cruciform specimen is an effective material test method to accurately detect andmodel the deformation behavior of sheet metals under biaxial tension, and consequently, to improve the predictive accuracyof FEA (Kuwabara et al., 2004, 2011; Hashimoto et al., 2010; Moriya et al., 2010; Yanaga et al., 2012). However, one of the draw-backs of this test method is that the maximum plastic strain applicable to a cruciform specimen is only several percent.

Ishiki et al. (2011) and Sumita et al. (2011) investigated the work hardening behavior of pure titanium sheet under biaxialtension. The maximum plastic strain applied to the test material was only 0.002 for the biaxial tensile tests using cruciformspecimens. Thus, tubular specimens were fabricated to successfully measure the biaxial work hardening behavior of thesheet sample for a larger strain range. The sheet material was bent into a cylindrical shape and the sheet edges were la-ser-welded together to apply combined internal pressure and tension to the tubular specimens using a servo-controlled tubebulge testing machine developed by Kuwabara et al. (2003, 2005). Consequently, the work hardening behavior of pure tita-nium sheet was successfully measured for many linear stress/strain paths up to a maximum plastic strain of 0.085 (Ishikiet al., 2011) and 0.225 (Sumita et al., 2011). However, biaxial stress–strain curves of the test material could not be contin-uously measured from initial yield up to fracture, due to detachment of the strain gauges.

The objective of the present study is to establish a material test method that enables the continuous measurement ofbiaxial stress–strain curves for sheet metals under biaxial tension. A servo-controlled, combined tension-internal pressuretesting machine for tubular specimens fabricated from flat sheet was developed, in addition to a strain measurement systemthat is capable of continuously measuring the axial and circumferential strain components and bulge curvature of a tubularspecimen from initial yield up to fracture. The work hardening behavior and forming limit strains and stresses for cold-rolledinterstitial-free (IF) steel sheet were successfully measured in detail for many linear stress paths to demonstrate the useful-ness of the multiaxial tube expansion testing machine. Furthermore, the most appropriate yield function for the test materialwas determined to calculate the forming limit strains and stresses by the Marciniak–Kuczynski approach (Marciniak andKuczynski, 1967). The calculated results were compared with the experimental results to discuss the validity of the Marci-niak–Kuczynski approach and the methodology for material modeling.

2. Development of multiaxial tube expansion testing machine

Fig. 1 shows a schematic diagram of the servo-controlled multiaxial tube expansion testing machine developed in thisstudy. An axial load T, and internal pressure P, were applied to a tubular specimen by a hydraulic cylinder and pressure

T. Kuwabara, F. Sugawara / International Journal of Plasticity 45 (2013) 103–118 105

booster, respectively. These were measured using a load cell and pressure gauge, respectively, and the variable ranges were�200 6 T 6 200 kN and 0 6 P 6 50 MPa.

Fig. 2 shows a schematic diagram of the strain measurement apparatus developed and installed to the testing machine toenable continuous measurement of the deformation of a tubular specimen.

Fig. 2(a) shows the sensor used for measurement of the circumferential strain esh, of a specimen. The average radial dis-

placement Dr, of the outer surface at the mid-section of a bulging specimen was measured using three displacement trans-ducers, DT1, DT2 and DT3; DT1 is located at a position of h = 180� from the weld line, and DT2 and DT3 are at positions ofh = ±60�. es

h is determined according to:

esh ¼ ln

D0 þ 2DrD0

; ð1Þ

where D0 is the initial outer diameter of the specimen. The current outer diameter D at the mid-section is determined from:

D ¼ D0 expðeshÞ: ð2Þ

Fig. 2(b) shows a schematic diagram of the sensor used for measurement of the radius of axial curvature R/, at the bulgingarea. Three displacement transducers, DT1A, DT1 and DT1B, were placed at intervals of s = 13 mm along the axial direction ofthe specimen. The difference of height h, was measured between DT1 and the average of DT1A and DT1B, and R/ was thencalculated using:

R/ ¼h2 þ s2

2h: ð3Þ

Fig. 2(c) shows a schematic diagram of the sensor used for measurement of the axial strain es/, at the mid-section of the

bulging specimen. The displacement transducer (DTEL) is used to detect the change in the distance between the contactwedges. es

/ was calculated according to:

es/ ¼ ln

2R/

L0sin�1 L

2R/

� �� ; ð4Þ

where L0 and L are the initial and current gauge lengths, respectively. L0 was set to 20 mm for the purposes of this study.From the assumption of a constant volume, the current wall thickness t, at the mid-section of the bulging specimen was

calculated using:

t ¼ D2� D

2

� �2

� ðD0 � t0Þt0

expð�es/Þ

( )1=2

; ð5Þ

where t0 is the initial wall thickness of the specimen.From Eqs. (1)–(5), the circumferential and axial strains, e/ and eh, at the mid wall of the specimen were evaluated using

the following equations:

eh ¼ lnD0 expðes

hÞ � tD0 � t0

e/ ¼ es/ � ln

R/

R/ � t=2

: ð6Þ

The axial and circumferential stresses, r/ and rh, at the mid-section of the bulging specimen were calculated as the valuesat the mid wall using the following equations:

r/ ¼PpðD=2� tÞ2 þ T

pðD� tÞt

rh ¼ðR/ � tÞðD� 2tÞð2R/ � tÞt P � D� t

2R/ � tr/

ð7Þ

based on the equilibrium requirements for a material element at the mid-section of a specimen.

Load cell

Hydraulic cylinder

Pressure booster

Pressure gauge

Tubular specimen

Hydraulic cylinder

Strain measurementapparatus

Fig. 1. Schematic diagram of the multiaxial tube expansion testing machine.

DT1

DT3 DT2

DT1 TD1BDT1A

120°

Measuringstick

2s

Weld line

θ

DT2

120°

60° Weld line

h

(a) (b)

Displacement transducer (DTEL)

Tubular specimenContact wedge

(c)

Fig. 2. Schematic diagram of the novel strain measurement apparatus. Sensors for measuring (a) circumferential strain eh, (b) the radius of curvature in theaxial direction, R/, and (c) axial strain, e/.

Fig. 3. Electrical feedback circuit for servo-control of the true stress paths applied to a tubular specimen.


Fig. 3 shows the feedback control circuit used for true stress control with the multiaxial tube expansion testing ma-chine. The measured values were input to calculate r/ and rh from Eq. (7). The calculated values were then comparedwith the command values for r/ and rh. From the discrepancy between the calculated and command values, T and Pwere controlled at an interval of 0.9 ms by servo-controlled hydraulic pumps installed to the testing machine. The res-olution in the measurement of T and P was 12.5 N and 6.25 kPa, respectively. The resolution in the measurement of dis-placement using the transducers shown in Fig. 2 was 1.56 lm The measured values were recorded onto a hard diskdrive every 5 ms.


3. Experimental procedure

3.1. Test material and specimens

The test material used in this study was 0.7 mm thick cold-rolled IF steel sheet. The work hardening characteristics and r-values at 0�, 45� and 90� (transverse direction; TD) to the rolling direction (RD) are listed in Table 1. Hereafter, the RD, TD,and the thickness direction of the material are defined as the x-, y-, and z-axes, respectively.

Two types of biaxial tensile tests were performed to measure the plastic deformation behavior of the test material frominitial yield up to fracture. Fig. 4(a) shows a schematic of the cruciform specimen used for the biaxial tensile test of the as-received sheet sample. The geometry of the specimen was the same as that proposed by Kuwabara et al. (1998). The spec-imen arms were parallel to the RD and TD of the material. Each arm of the specimen had seven slits, 60 mm long and 0.2 mmwide, at 7.5 mm intervals, to exclude geometric constraint on the deformation of the 60 � 60 mm2 square gauge area. Theslits were fabricated by laser cutting.

Normal strain components (ex, ey) were measured using uniaxial strain gauges (YFLA-2, Tokyo Sokki Kenkyujo Co.)mounted at ±21 mm from the center along the maximum loading direction. According to a FEA of the cruciform specimenwith the strain measurement position shown in Fig. 4(a), the stress measurement error was estimated to be less than 2%(Hanabusa et al., 2010; Hanabusa et al., 2013). Details of the biaxial tensile testing apparatus and test method are givenin Kuwabara et al. (1998, 2000).

Fig. 4(b) shows a schematic of the tubular specimen used for the multiaxial tube expansion tests. The specimens werefabricated by bending the sheet sample into a cylindrical shape and laser-welding the sheet edges together to produce tubu-lar specimens withan outer diameter of 46.0 mm, a length of 200 mm and a gauge length (distance between the grips of thetesting machine) of 140 mm. The width of the weld line, including that of heat affected zones, measured approximately1 mm based on the data of Vickers hardness distribution across the weld line. Two types of tubular specimens were fabri-cated; type I specimen had the RD in the axial direction and type II specimen had the RD in the circumferential direction.Type I specimens were used for tests with rx < ry and type II for tests with rx P ry; the maximum principal stress directionwas always taken to be in the circumferential direction.

3.2. Hydraulic bulge test

As mentioned later in Section 4.4, the maximum value of ep0 attained using a tubular specimen for equibiaxial tension,

rx:ry = 1:1, was 0.15 because of fracture occurring in the weld line. Therefore, the hydraulic bulge test was performed tomeasure the work hardening behavior of the material for a strain range of ep

0 > 0:15. Fig. 5 shows a schematic overviewof the hydraulic bulge test. The diameter of the die opening was 150 mm, the die profile radius was 8 mm, and the blankdiameter was 220 mm. The material flow-in was fixed at zero along the boundary of a 190 mm diameter using a triangulardraw-bead. No lubricant was used at the interface between the blank and die surface.

The true thickness strain component ez, and the radius of curvature q, at the top of the bulged specimen were measuredusing an optical deformation measurement system (ARAMIS�, GOM) during forming. The gauge length was a diameter of30 mm. The hydraulic pressure P, was controlled to maintain the equivalent plastic strain rate as approximately constant(ca. 5 � 10�4 s�1) during each test. The measured data of ez, q and P were recorded onto a hard disk every 0.1 s using a datalogger.

The in-plane equibiaxial stress rb, at the apex of the bulged specimen was measured as:

rb ¼Pq2t; ð8Þ

where t is the thickness at the apex of the bulged specimen, as determined using:

t ¼ t0 expðezÞ ¼ t0 expð�ex � eyÞ; ð9Þ

based on the condition of constant volume, in which the elastic strain components were neglected.

Table 1Mechanical properties of the test material.

Tensile direction r0.2 (MPa) c (MPa) n a r-valuec

0� 164 574a 0.273a 0.008a 2.270� – 556b 0.245b �0.004b –45� 173 574a 0.272a 0.008a 1.7745� – 548b 0.212b �0.012b

90� 170 564a 0.273a 0.009a 2.6590� – 540b 0.234b �0.007b –

a Approximated using r = c(a + ep)n for 0:002 6 ep6 0:093.

b Approximated using r = c(a + ep)n for 0:05 < ep6 0:24.

c Measured at nominal strain eN = 0.1.

(a)

L140

type II

φ 46

0.7

200

Clamping area

type I

(b)

Rolling direction

60

60

x

y

R1

Slit width: 0.2 Strain gauges

7.5

260

21 21260

60

Fig. 4. Schematic diagrams of specimens used for the biaxial tensile tests (dimensions in mm): (a) cruciform specimen, and (b) tubular specimen. M: rollingdirection. The strain gauge position shown in (a) is for the case that the maximum loading direction is parallel to the rolling direction of the sheet sample.

150

Spherometer

P

190

150

R8

CCDCCD

CCD

CCD

Fig. 5. Schematic overview of the hydraulic bulge test.



3.3. Measurement of contours of plastic work

Both cruciform and tubular specimens were subjected to proportional loading with true stress ratios rx:ry = 4:1, 2:1, 4:3,1:1, 3:4, 1:2 and 1:4. A standard uniaxial tensile specimen (JIS 13 B-type) was used for the uniaxial tensile tests withrx:ry = 1:0 and 0:1. True stress increments were controlled and applied to the specimens so that the von Mises equivalentplastic strain rate became approximately constant at 5 � 10�4 s�1 for all stress paths. Two specimens were used for eachstress ratio.

The concept of the contour of plastic work in the stress space (Hill and Hutchinson, 1992; Hill et al., 1994) was used toquantitatively evaluate the work hardening behavior of the test material under biaxial tension. The stress–strain curve ob-tained from a uniaxial tensile test in the RD was selected as a reference datum for work hardening; the uniaxial tensile truestress r0, and the plastic work per unit volume W0, associated with a particular value of offset true plastic strain ep

0, weredetermined. The uniaxial true stress r90, in the TD and the biaxial true stress components (rx,ry) were then determinedat the same plastic work as W0. The stress points (r0,0), (0,r90) and (rx,ry) thus plotted in the principal stress space forma plastic work contour associated with a particular value of ep

0. When ep0 is taken as sufficiently small, the work contour can be

practically viewed as a yield locus.

4. Results and discussion

4.1. True stress–true plastic strain curves

Fig. 6 shows true stress-true plastic strain curves (s–s curves) measured using the tubular specimens. The s–s curves weresuccessfully measured up to a strain level of specimen fracture for all stress ratios. The s–s curves measured using cruciformspecimens are also depicted in each figure for all stress ratios.

Slight differences in flow stress between the tubular and cruciform specimens are observed for all stress ratios, due to theinfluence of the prestrain applied to the sheet sample during tube fabrication. The prestrain distributes linearly in the thick-ness direction, where it is 0 at the mid thickness and takes the maximum and minimum values, ±t0/(D0 � t0), at the outer andinner surfaces of the tube, respectively; these were ±0.015 for the tube geometry shown in Fig. 4(b). The influence of pre-strain on the s–s curves measured in the multiaxial tube expansion tests is diminished with increase of the subsequent strainapplied to the tubular specimen during the tube expansion tests. However, it cannot be neglected during the initial defor-mation stages in the tube expansion tests, due to the memory effect of prestrained metals (Miastkowski and Szczepinski,1965; Williams and Svensson, 1970; Hecker, 1971).

From this reason, the s–s curves measured for the multiaxial tube expansion tests were corrected to appropriately reflectthe influence of prestrain applied to the tubular specimens as follows:

1. The s–s curves measured using cruciform specimens are used as reference data.2. Shift the s–s curves for the x- and y-directions measured using tubular specimens along the strain axis to find a connect-

ing point, q, at which both s–s curves smoothly connect to those measured using a cruciform specimen for the samestress ratio.

Consequently, the s–s curves measured using tubular specimens are smoothly connected to those measured using cruci-form specimens for all stress ratios. The maximum and minimum amounts of s–s curve shift in the strain axis were 0.004 and�0.002, respectively. The biaxial s–s curves measured using cruciform specimens were henceforth adopted as the experi-mental data for the strain range lower than the connecting point, and those measured using tubular specimens were adoptedas the experimental data for the strain range higher than the connecting point.

Fig. 6(h) shows a comparison of the s–s curve measured using a tubular specimen for rx:ry = 1:0 with that measuredusing a JIS 13-B type uniaxial tensile specimen in the RD. The former was successfully measured for a strain range ofep

0 6 0:21. Both curves are in good agreement with each other for a strain range of 0.063 (connecting point) 6 ep0 6 0:21,

which verifies the validity of the offset shift method.

4.2. Contours of plastic work

Fig. 7(a) shows measured stress points forming the contours of plastic work for different levels of ep0. Each stress point

represents an average of two specimen data; the difference of the two measured data was less than 1% of the flow stressfor all data points. The maximum value of ep

0 for which the work contour has a full set of nine stress points wasep

0 ¼ 0:36, which is approximately ten times larger than that obtained using a cruciform specimen. Moreover, it is notewor-thy that the maximum strain of ep

0 ¼ 0:50 was attained for rx:ry = 1:4 and 1:1.Fig. 7(b) compares the work contours shown in Fig. 7(a) with those determined using the as-measured biaxial s–s curves

(with no shift along the strain axis). It is clear that both work contours almost coincide with each other for a strain rang ofep

0 P 0:035. Therefore, it is concluded that shifting s–s curves along the strain axis is not necessary if the objective of themultiaxial tube expansion test is to investigate the biaxial deformation behavior of the test material for a large strain range.

-0.02 0.00 0.02 0.04 0.060

100

200

300

400

(a)σy

σx

σx : σy = 4 : 1

True

stre

ss σ

/ M

Pa

True plastic strain ε p-0.02 0.00 0.02 0.04 0.060

100

200

300

400

(b)

σy

σxσx : σy = 2 : 1


100

200

300

400

(c)

σy

σx

σx : σy = 4 : 3

True plastic strain ε p

-0.02 0.00 0.02 0.04 0.060

100

200

300

400

(d)

σy

σx

σx : σy = 1 : 4

True

stre

ss σ

/ M

Pa


100

200

300

400

(e)

σx

σyσx : σy = 1 : 2


100

200

300

400

(f)

σy

σxσx : σy = 3 : 4


-0.02 0.00 0.02 0.04 0.060

100

200

300

400

(g)

σx

σx : σy = 1 : 1

True

stre

ss σ

/ M

Pa

True plastic strain ε p0.00 0.05 0.10 0.15 0.20

0

100

200

300

400

(h)

σx

Tubular specimen JIS 13-B specimen

σx : σy = 1 : 0


Tubular specimen Cruciform specimen Connecting point

Fig. 6. Offset of the true stress-true plastic strain curves for rx:ry = (a) 4:1, (b) 2:1, (c) 4:3, (d) 1:4, (e) 1:2, (f) 3:4, (g) 1:4, and (h) 1:0 measured using themultiaxial tube expansion testing machine.


The stress values shown in Fig. 7(c) form the work contours for ep0 ¼ 0:005, 0.15 and 0.36 shown in Fig. 7(a), normalized by

r0 associated with the respective work contours. The shapes of the work contours are significantly changed with work hard-ening, or equivalently with ep

0, i.e., the test material exhibited differential work hardening (Hill and Hutchinson, 1992; Hillet al., 1994). Table 2 shows the measured stress values forming the work contours for ep

0 ¼ 0:005, 0.15 and 0.36.The theoretical yield loci based on the von Mises (Von Mises, 1913), Hill’s quadratic (Hill, 1948) and the Yld2000-2d yield

function (Barlat et al., 2003; Yoon et al., 2004) with an exponent of M = 6 determined for the ep0 ¼ 0:005, 0.15 and 0.36 work

contours are superimposed in Fig. 7(c). The unknown parameters of the Hill’s quadratic yield function were determinedusing r0; r45; r90 and r0/r0, and those of the Yld2000-2d yield function were determined using r0; r45; r90 and rb and r0/r0, r45/r0, r90/r0 and rb/r0, where r# and r# are the r-value and tensile flow stress measured at an angle # from the RD,respectively, and where rb and rb are the ratio of the plastic strain rates, dep

y=depx , and the flow stress at equibiaxial tension,

rx:ry = 1:1, respectively. The values of r0, r45 and r90 used to determine the Hill ‘48 and Yld2000-2d yield functions were thesame as those in Table 1, and the values of rb and rb used to determine the Yld2000-2d yield function are shown in Table 2.The values of the unknown a1 to a8 parameters determined for the respective Yld2000-2d yield functions are given inTable 3.

Furthermore, the standard deviations dr, of the theoretical yield loci from the measured work contours were calculated fora quantitative evaluation of the most suitable yield function for the test material (see Appendix A). Fig. 8 shows the calcu-lated results for dr. The Yld2000-2d yield function with M = 6 provides closer agreement with the experimental data than theother yield functions for almost all levels of ep

0.The shape ratios of the work contours were determined and are shown in Fig. 9 for a quantitative evaluation of the amount

of differential work hardening. The shape ratio is defined as a/a0, where a0 is the distance between the origin in the principalstress space and a stress point that forms the work contour for ep

0 ¼ 0:005, and a is the distance between the origin in the

0

100

200

300

400

500

600

σ y /

MPa

σx / MPa

Hydraulic bulge test with shift ε p

0 0.005 0.015 0.035 0.075 0.15 0.25 0.36 0.46 0.50

(a)

0

100

200

300

400

500

600

σ y /

MPa

σx / MPa

with withshift no shift ε p

0 0.002 0.005 0.015 0.035 0.075 0.15 0.25 0.36 0.46 0.50

(b)

0 100 200 300 400 500 600

0 100 200 300 400 500 600

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5Hill '48

von Mises

Yld2000-2d(M = 6) ε p

00.0050.150.36

σ y /

σ 0

σx / σ

0

(c)

Fig. 7. Measured stress points forming contours of plastic work. In (a), the work contours are determined using the biaxial stress–strain curves with shiftsalong the strain axis, as shown in Fig. 6. In (b), the work contours shown in (a) are compared with those based on the as-measured biaxial stress–straincurves (with no shift along the strain axis). In (c), the stress values corresponding to a specific value of ep

0 are normalized by r0 associated with the samegroup of work contour.


principal stress space and a stress point that forms a work contour for a particular value of ep0 (>0.005). (a/a0) > 1 and

(a/a0) < 1 indicate the expansion and shrinkage of the work contour, respectively. Comparison of the shape ratios forrx:ry = 4:1, 2:1, 4:3 (rx > ry) and rx:ry = 1:4, 1:2, 3:4 (rx < ry) indicates that both exhibit expansion; however, the degreeof expansion is larger for the former than for the latter. A similar tendency was observed for steel sheets with an averager-value larger than 1.5 (Kuwabara et al., 1998, 2002; Hora et al., 2009); therefore, the differential work hardening behavior

Table 2Measured stress values forming work contours.

ep0

0.005 0.15 0.36

r0 (MPa) 175.1 346.4 431.1rb 0.98 0.99 0.92

rx:ry rx/r0 ry/r0 rx/r0 ry/r0 rx/r0 ry/r0

1:0 1.000 0.000 1.000 0.000 1.000 0.0004:1 1.132 0.284 1.195 0.299 1.163 0.2902:1 1.216 0.609 1.272 0.637 1.232 0.6174:3 1.190 0.893 1.252 0.939 1.216 0.9131:1 1.081 1.083 1.162 1.162 1.141 1.1413:4 0.908 1.210 0.938 1.248 0.920 1.2301:2 0.614 1.227 0.627 1.253 0.613 1.2261:4 0.284 1.134 0.297 1.180 0.290 1.1570:1 0.000 1.019 0.000 0.988 0.000 0.982

Table 3Values of unknown parameters for the Yld2000-2d yield function shown in Fig. 7(c).

ep0

a1 a2 a3 a4 a5 a6 a7 a8

0.005 1.0718 1.0322 0.9764 0.9078 0.9171 0.9189 1.0041 0.89370.15 0.9959 1.1255 0.8334 0.8866 0.9015 0.7519 1.0038 1.00440.36 0.9919 1.1392 0.8499 0.8917 0.9118 0.8024 1.0235 1.0251

0.0 0.1 0.2 0.3 0.40.00

0.05

0.10

0.15

0.20

0.25

0.30 von Mises Hill '48

Yld2000-2dM = 4M = 5M = 6

Stan

dard

dev

iatio

n δ r

Reference plastic strain ε p 0

Fig. 8. Standard deviation dr of the calculated yield loci from the measured work contours.

0.0 0.1 0.2 0.3 0.4 0.5 0.60.90

0.95

1.00

1.05

1.10σ

x : σ

y

4 : 1 2 : 1 4 : 3 1 : 1 3 : 4 1 : 2 1 : 4 0 : 1Sh

ape

ratio

a /

a 0

Reference plastic strain ε p

0

0

σy /σ0

σx /σ0

a0

Referencework contour

p0( 0.005)ε =

σx :σy

a

Fig. 9. Variation of the shape ratio a/a0, with ep0 for respective linear stress paths.


shown in Fig. 9 is deemed to be a common feature for steel sheets with relatively high r-values. The a/a0 for rx:ry = 0:1 de-creases steeply up to ep

0 � 0:05 and gradually decreases for ep0 > 0:05, while a/a0 for other stress ratios increases steeply up to

ep0 � 0:05 (0.15 for rx:ry = 1:1 and 3:4) and gradually decreases with an increase of ep

0.

0.0 0.1 0.2 0.3 0.4 0.5 0.6-30

0

30

60

90

120σ

x : σ

y

4 : 1 2 : 1 4 : 3 1 : 1 3 : 4 1 : 2 1 : 4

Dire

ctio

n of

pla

stic

stra

in ra

te β

/ °

Reference plastic strain ε p

00

σy

σx

β

Fig. 10. Variation of the direction of plastic strain rates with ep0 for respective linear stress paths.


4.3. Directions of plastic strain rates

Fig. 10 shows the variation in the directions of plastic strain rates b, with ep0 measured for each stress path. b is defined as

0 when it is parallel to the RD, and the increment of b in the anticlockwise direction is positive. b is almost constant for rx:-ry = 2:1, 1:1 and 1:2. However, b gradually decreases for rx:ry = 1:4 and 4:3 and increases for rx:ry = 4:1 and 3:4 with anincrease of ep

0; the amount of increment Db, from ep0 = 0.005 to 0.36 was �1� and �6.5� for rx:ry = 1:4 and 4:3, respectively,

and +3� and +6� for rx:ry = 4:1 and 3:4, respectively.To confirm if the measured work contours could be considered to be a plastic potential, the measured data of b were com-

pared with the directions of outward vectors normal to the selected theoretical yield loci. The result is shown in Fig. 11 for ep0

= 0.005, 0.15 and 0.36. Furthermore, the standard deviations of the calculated directions of plastic strain rates from thosemeasured db (see Appendix A) are shown in Fig. 12. The Yld2000-2d yield function with M = 6 provides a closer agreementwith the experimental data than the other yield functions for almost all levels of ep

0, where db < 4� except for ep0 ¼ 0:05.

From the results of Figs. 8 and 12, it is concluded that the Yld2000-2d yield function with M = 6 is the most appropriatematerial model for the test material, at least for linear stress paths. Consequently, the measured biaxial stresses and plasticstrain rates observed for the linear stress paths lead to the optimum choice of exponent M for the Yld2000-2d yield function.The biaxial tensile test methods using cruciform specimens and tubular specimens are thus verified as an effective materialtest method to accurately detect and model the deformation behavior of sheet metals under biaxial tension.

4.4. Forming limit strains and stresses

Fig. 13(a) shows fracture specimens for respective linear stress paths. Localized necks appeared all around the centralbulged area of the respective specimens for all stress ratios, except for rx:ry = 1:1, as shown in Fig. 13(b) for rx:ry = 1:2and Fig. 13(c) for rx:ry = 1:4.

Fig. 14(a) shows the forming limit strains of the test material. Fig. 14(b) shows the forming limit stresses that are definedas the measured true stress components (rx, ry) at the instant when the specimen reached the forming limit strain. Theforming limit strains for the multiaxial tube expansion tests were determined as the logarithmic plastic strains, ep

x and epy ,

at the moment immediately before a sudden decrease of internal pressure following specimen fracture. epx and ep

y were cal-culated by subtracting elastic strains from the total strains measured using Eq. (6). It is noted that the specimens thatreached the forming limit strain had slight localized necks as shown in Fig. 13(b) and (c); therefore, the measured forminglimit strains are possibly somewhat larger than those at the onset of strain localization.

The forming limit strains for rx:ry = 1:1, 1:0 and 0:1 could not be measured using tubular specimens, because a smallcrack occurred in the weld line for rx:ry = 1:1, and axial buckling occurred at the tube ends near the grips of the testing ma-chine for rx:ry = 1:0 and 0:1. Therefore, hydraulic bulge tests and conventional uniaxial tensile tests with a JIS 13B-typespecimen (gauge length: 60 mm, specimen width: 12.5 mm) were performed to measure the forming limit strains and stres-ses for rx:ry = 1:1, 1:0 and 0:1. For the hydraulic bulge test, the forming limit strains were determined as ep

x ¼ epy ¼ �ep

z =2,where ep

z is the logarithmic plastic strain in thickness direction measured using a micrometer, in close vicinity of fractureposition occurring at the top of the bulged specimen. For the uniaxial tensile test, ARAMIS� was used to determine the thick-ness strain distribution in the gauge area at the onset of a localized neck that appeared in a defused necking area; the form-ing limit strain was determined as the strain value in the area right next to the localized neck. The measured forming limitstrains and stresses for the hydraulic bulge test and uniaxial tensile tests are also included in Fig. 14(a) and (b).

The symbols h and w in Fig. 14(a) and (b) indicate that fracture occurred at a position of h 6 ±30� and h > ±30�, respec-tively, where h is the angle from the weld line in the circumferential direction of a tubular specimen (see Fig. 2(a)). Therefore,w is deemed to represent the real forming limit strains and stresses of the test material, because the position of fracture is

0 15 30 45 60 75 90-45

0

45

90

135

Hill '48

von Mises Yld2000-2d(M = 6) ε p

00.0050.150.36

ϕ

σ

σ x

y

β

Dire

ctio

n of

pla

stic

stra

in ra

te β

/ °

Loading direction ϕ / °

Fig. 11. Measured directions of plastic strain rates compared with those calculated using selected yield functions.

0.0 0.1 0.2 0.3 0.40

5

10

15 von Mises Hill '48

Yld2000-2dM = 4M = 5M = 6

Stan

dard

dev

iatio

n δ β

/°

Reference plastic strain ε p 0

Fig. 12. Standard deviation db, of the directions of plastic strain rates calculated using selected yield functions from those measured.


sufficiently distant from the weld line, and therefore, the weld line does not appear to have any effect on the occurrence offracture.

For rx:ry = 4:1, 2:1, 4:3 and 1:4, specimens fractured at both h 6 ±30� and h > ±30�. It is conjectured that the scatter offracture position was caused by subtle scatter of fabrication conditions for the tubular specimens. However, it should benoted that h are always close to w for these stress ratios; the differences in the amount of strain between h and w for rx:-ry = 4:1, 2:1, 4:3 and 1:4 were 0.05, 0.01, 0.02 and 0.04, respectively, and were therefore deemed to be almost within therange of experimental scatter in the strain measurement. Therefore, h can also be viewed as the real forming limit strainsand stresses of the test material for the given stress ratios.

For rx:ry = 3:4 and 1:2, fracture always occurred at a position of h 6 ±30�, 5 to 10 mm apart from the weld line, for allspecimens (4-5 specimens). However, the symbol h for these stress ratios is deemed to represent the real forming limitstrains and stresses of the test material, because slight localized necks were observed all around the central bulged areaof the respective specimens.

Thus, multiaxial tube expansion testing in addition to hydraulic bulge and uniaxial tensile tests allows the forming limitstrains and stresses under linear stress paths to be fully determined in the first quadrant of the stress space.

Fig. 14(a) and (b) also include the forming limit curve (FLC) and forming limit stress curve (FLSC) calculated using theMarciniak–Kuczynski approach (Marciniak and Kuczynski, 1967). The yield functions used were the respective Yld2000-2d yield functions with an exponent of M = 6 determined for ep

0 = 0.005, 0.15 and 0.36, as shown in Fig. 7(c). Isotropic hard-ening was assumed for the analysis. The hardening curve used in the analysis was r ¼ 574ð0:008þ epÞ0:273 (MPa) in the RD,based on the curve measured using an extensometer for 0:002 6 ep

x 6 0:093, see Table 1. An additional test confirmed thatthe hardening curve was in fair agreement with that measured using ARAMIS� for 0:002 6 ep

x 6 0:55. The magnitude of ini-tial imperfection, the strain rate sensitivity exponent (m-value), and the equivalent plastic strain rate were designated as0.994, 0.02, and 0.0005 s�1, respectively. Please refer to Yoshida et al. (2007) for details of the calculation procedures. TheFLC and FLSC calculated using the Yld2000-2d yield function determined for ep

0 = 0.15 and 0.36 are in fair agreement withthe experimental data, which validates the Marciniak–Kuczynski approach.

2:1 4:14:31:14:1 2:1 4:3:x yσ σ

(a)

(b) (c)

Fig. 13. (a) Fracture specimens for respective linear stress paths. Localized necks observed for (b) rx:ry = 1:2 and (c) rx:ry = 1:4; black paint was applied tothe localized necks for good visibility.

(a) (b)Fig. 14. Measured forming limit (a) strains and (b) stresses, compared with the forming limit curve (FLC) and forming limit stress curve (FLSC), respectively,calculated using the Marciniak–Kuczynski approach.


The reason why the Yld2000-2d yield function determined for ep0 = 0.15 and 0.36 gives lower limit strains in biaxial ten-

sion than that determined for ep0 ¼ 0:005 is that the former gives a more ‘‘stretched out’’ yield locus along the equibiaxial

tension axis, rx/ry = 1, than the latter (Sowerby and Duncan, 1971), as shown in Fig. 7(c). Nevertheless, the limit stressescalculated using the Yld2000-2d yield function determined for ep

0 = 0.005, 0.15 and 0.36 are almost identical with one an-other in biaxial tension. This is possibly because the decrease in limit stresses caused by the neglect of differential workhardening in the FLC analysis using the Yld2000-2d yield function for ep

0 ¼ 0:005 cancelled out the increase in limit stressesdue to the increase in limit strains.


5. Conclusions

A servo-controlled multiaxial tube expansion testing machine that enables the measurement of biaxial stress–straincurves for sheet metals from initial yield up to fracture was developed. To demonstrate the usefulness of the testing machine,biaxial stress tests for cold-rolled IF steel sheet were performed. The conclusions of this study are summarized as follows:

1. A novel strain measurement apparatus that enables continuous measurement of the biaxial strain components, in addi-tion to the radius of curvature in the axial direction of a bulging tubular specimen from initial yield up to fracture, wasdeveloped for installation with the multiaxial tube expansion testing machine (Fig. 2). Biaxial stress–strain curves mea-sured for tubular specimens fabricated from the as-received sheet sample were consistently connected with those mea-sured for cruciform specimens for all the linear stress paths tested (Fig. 6). Thus, the accuracy of the new strainmeasurement apparatus was verified.

2. Contours of plastic work for the test material were successfully measured for an equivalent plastic strain range of 0.0056 ep

0 6 0:36 (Fig. 7). The test material exhibited significant differential work hardening; the shapes of the work contourswere significantly changed with work hardening, or equivalently with ep

0 (Fig. 9). The Yld2000-2d yield function with anexponent of M = 6 provided the closest agreement with the measured work contours (Fig. 8) and directions of plasticstrain rates (Fig. 12) for a strain range of 0.005 6 ep

0 6 0:36.3. The forming limit curve (FLC) and forming limit stress curve (FLSC) of the test material subjected to linear stress paths

were successfully measured for the first time (Fig. 14(a) and (b)). The measured FLC and FLSC were in fair agreement withthose calculated using the Marciniak–Kuczynski approach with an assumption of isotropic hardening.

4. It was clearly established that the multiaxial tube expansion test is effective to measure the multiaxial deformationbehavior of sheet metals for a large strain range. However, a concurrent use of the biaxial tensile test using a cruciformspecimen is necessary to accurately measure multiaxial stress–strain curves for a small strain range (Fig. 6).

Acknowledgements

The authors wish to acknowledge assistance and advice received from Mr. Shigeru Matsumoto, Mr. Kazuyoshi Tashiro andMr. Akio Masaki of Kokusai Co., Ltd. (Japan), regarding the fabrication of the testing machine. Provision of the test materialfrom Sumitomo Metal Industries, Ltd. is appreciated. T. K. expresses his thanks to the members of the Committee for Stan-dardizing Steel Tube Formability Evaluation Methods organized in The Iron and Steel Institute in Japan and Dr. Thomas B.Stoughton of General Motors for helpful suggestions and discussions. This work was partly supported by funding fromthe Ministry of Economy, Trade and Industry of Japan, and the Ministry of Education, Culture, Sports, Science and Technologyof Japan (KAKENHI, 23360318).

Appendix A

To quantitatively compare the difference between the shapes of the theoretical yield loci and the measured work con-tours, the standard deviation dr, was calculated using the following equation:

Fig. A1deviatio

dr ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPifr0ðuiÞ � rðuiÞg

2

N � 1

s; ðA1Þ

where ui (i = 1 to N) is the loading angle of the ith stress point from the x-axis in the principal stress space, r(ui) is the dis-tance between the origin of the principal stress space and the ith stress point, and r0(ui) is the distance between the origin ofthe principal stress space and the theoretical yield locus along the loading direction ui, as shown by the schematic inFig. A1(a).

σx /

σ y/

iϕ

ir

ir

σ0

σ0

x /

y/

i

ir

ir

0

0

ϕ

β

iϕ

iβiβ

(a) (b) . Schematics for calculation of (a) the standard deviation dr, of the calculated yield loci from the measured plastic work contours, and (b) the standardn db, of the calculated directions of plastic strain rates based on the normality flow rule for respective theoretical yield loci from those measured.


To quantitatively evaluate the difference between the measured directions of plastic strain rates and those predictedusing the selected yield functions, the standard deviation db, was calculated using the following equation:

db ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPifb0ðuiÞ � bðuiÞg

2

N � 1

s; ðA2Þ

where b(ui) is the direction of plastic strain rate measured for the ith stress path, and b0(ui) is that predicted using a selectedyield function for the ith stress path, as shown by the schematic in Fig. A1(b).

References

An, Y., Vegter, H., Carless, L., Lambriks, M., 2011. A novel yield locus description by combining the Taylor and the relaxed Taylor theory for sheet steels. Int. J.Plast. 27, 1758–1780.

Andar, M.O., Kuwabara, T., Steglich, D., 2012. Material modeling of AZ31 Mg sheet considering variation of r-values and asymmetry of the yield locus. Mater.Sci. Eng. A 549, 82–92.

Banabic, D., Barlat, F., Cazacu, O., Kuwabara, T., 2010. Advances in anisotropy and formability. Int. J. Mater. Form. 3, 165–189.Bramley, A.N., Mellor, P.B., 1966. Plastic flow in stabilized sheet steel. Int. J. Mech. Sci. 8 (2), 101–114.Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi, S.H., Chu, E., 2003. Plane stress yield function for aluminum alloy

sheets—Part 1: Theory. Int. J. Plast. 19, 1297–1319.Bell, R., Duncan, J.L., Wilson, I.H., 1967. A sheet-bulging machine with closed-loop control. J. Strain Anal. 2 (3), 246–253.Dunand, M., Maertens, A.P., Luo, M., Mohr, D., 2012. Experiments and modeling of anisotropic aluminum extrusions under multi-axial loading – Part I:

Plasticity. Int. J. Plast. 36, 34–49.Hanabusa, Y., Takizawa, H., Kuwabara, T., 2010. Evaluation of accuracy of stress measurements determined in biaxial stress tests with cruciform specimen

using numerical method. Steel Res. Int. 81 (9), 1376–1379.Hanabusa, Y., Takizawa, H., Kuwabara, T., 2013. Numerical verification of a biaxial tensile test method using a cruciform specimen. J. Mater. Processing

Technol. http://dx.doi.org/10.1016/j.jmatprotec.2012.12.007.Hashimoto, K., Kuwabara, T., Iizuka, E., Yoon, J.W., 2010. Effect of anisotropic yield functions on the accuracy of hole expansion simulations for 590 MPa

grade steel sheet. Tetsu-to-Hagané 96, 557–563 (in Japanese).Hecker, S.S., 1971. Yield surfaces in prestrained aluminum and copper. Metal. Trans. 2 (8), 2077–2086.Hecker, S.S., 1976. Experimental studies of yield phenomena in biaxially loaded metals. In: Stricklin, J.A., Saczalski, K.H. (Eds.), Constitutive Equations in

Viscoplasticity: Computational and Engineering Aspects. ASME, New York, pp. 1–33.Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. London A193 (1033), 281–297.Hill, R., Hecker, S.S., Stout, M.G., 1994. An investigation of plastic flow and differential work hardening in orthotropic brass tubes under fluid pressure and

axial load. Int. J. Solids Struct. 31, 2999–3021.Hill, R., Hutchinson, J.W., 1992. Differential hardening in sheet metal under biaxial loading: a theoretical framework. J. Appl. Mech. 59, S1–S9.Hora, P., Hochholdinger, B., Mutrux, A., Tong, L. 2009. Modeling of anisotropic hardening behaviour based on Barlat 2000 yield locus description, In: Hora, P.

(Ed.), Forming Technology Forum 2009, Zurich, pp. 21–29.Ikegami, K., 1979. Experimental plasticity on the anisotropy of metals. In: Boehler, J.P. (Ed.), Mechanical Behavior of Anisotropic Solids, Proceedings of the

Euromech Colloquim 115. Colloques Inter. du CNRS, Paris, pp. 201–242.Ishiki, M., Kuwabara, T., Hayashida, Y., 2011. Measurement and analysis of differential work hardening behavior of pure titanium sheet using spline

function. Int. J. Mater. Form. 4 (2), 193–204.Kelley, E.W., Hosford, W.F., 1968. The deformation characteristics of textured magnesium. Trans. TMS–AIME 242, 654–661.Khan, A.S., Wang, X., 1990. An experimental study of large finite plastic deformation in annealed 1100 aluminum during proportional and nonproportional

biaxial compression. Int. J. Plast. 6 (4), 485–504.Khan, A.S., Kazmi, R., Farrokh, B., 2007a. Multiaxial and non-proportional loading responses, anisotropy and modeling of Ti–6Al–4V titanium alloy over wide

ranges of strain rates and temperatures. Int. J. Plast. 23, 931–950.Khan, A.S., Chen, X., Abdel-Karim, M., 2007b. Cyclic multiaxial and shear finite deformation response of OFHC: Part I, experimental results. Int. J. Plast. 23,

1285–1306.Khan, A.S., Liu, H., 2012. A new approach for ductile fracture prediction on Al 2024-T351 alloy. Int. J. Plast. 35, 1–12.Korkolis, Y.P., Kyriakides, S., 2008a. Inflation and burst of anisotropic aluminum tubes for hydroforming applications. Int. J. Plast. 24, 509–543.Korkolis, Y.P., Kyriakides, S., 2008b. Inflation and burst of aluminum tubes. Part II: An advanced yield function including deformation-induced anisotropy.

Int. J. Plast. 24, 1625–1637.Korkolis, Y.P., Kyriakides, S., 2009. Path-dependent failure of inflated aluminum tubes. Int. J. Plast. 25, 2059–2080.Kuroda, M., Tvergaard, V., 1999. Use of abrupt strain path change for determining subsequent yield surface. Illustrations of basic idea. Acta Mater. 47, 3879–

3890.Kuwabara, T., Ikeda, S., Kuroda, T., 1998. Measurement and analysis of differential work-hardening in cold-rolled steel sheet under biaxial tension. J. Mater.

Process Technol. 80–81, 517–523.Kuwabara, T., Kuroda, M., Tvergaard, V., Nomura, K., 2000. Use of abrupt strain path change for determining subsequent yield surface. Experimental study

with metal sheets. Acta Mater. 48, 2071–2079.Kuwabara, T., Van Bael, A., Iizuka, E., 2002. Measurement and analysis of yield locus and work-hardening characteristics of steel sheets with different R-

values. Acta Mater. 50 (14), 3717–3729.Kuwabara, T., Ikeda, S., 2002. Measurement and analysis of work-hardening of sheet steels subjected to plane-strain tension. Tetsu-to-Hagané 88 (6), 334–

339 (in Japanese).Kuwabara, T., Ishiki, M., Kuroda, M., Takahashi, S., 2003. Yield locus and work-hardening behavior of a thin-walled steel tube subjected to combined

tension–internal pressure. J. de Phys. IV 105, 347–354.Kuwabara, T., Ikeda, S., Asano, Y., 2004. Effect of anisotropic yield functions on the accuracy of springback simulation. In: Ghosh, S. (Ed.), Proc. 8th Int. Conf.

on Numerical Methods in Industrial Forming Processes, American Institute of Physics, New York, p. 887.Kuwabara, T., Yoshida, K., Narihara, K., Takahashi, S., 2005. Anisotropic plastic deformation of extruded aluminum alloy tube under axial forces and internal

pressure. Int. J. Plast. 21, 101–117.Kuwabara, T., 2007. Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations. Int. J. Plast. 23, 385–

419.Kuwabara, T., Kumano, Y., Ziegelheim, J., Kurosaki, I., 2009. Tension–compression asymmetry of phosphor bronze for electronic parts and its effect on

bending behavior. Int. J. Plast. 25, 1759–1776.Kuwabara, T., Nakajima, T., 2011. Material modeling of 980 Mpa dual phase steel sheet based on biaxial tensile test and in-plane stress reversal test. J. Solid

Mech. Mater. Eng. 5, 709–720.Kuwabara, T., Hashimoto, K., Iizuka, E., Yoon, J.W., 2011. Effect of anisotropic yield functions on the accuracy of hole expansion simulations. J. Mater. Process.

Technol. 211, 475–481.

http://dx.doi.org/10.1016/j.jmatprotec.2012.12.007


Lee, D., Backofen, W.A., 1966. An experimental determination of the yield locus for titanium and titanium-alloy sheet. Trans. TMS–AIME 236, 1077–1084.Lowden, M.A.W., Hutchinson, W.B., 1975. Texture strengthening and strength differential in titanium-6A-4V. Metall. Trans. 6A, 441–448.Lou, X.Y., Li, M., Boger, R.K., Agnew, S.R., Wagoner, R.H., 2007. Hardening evolution of AZ31B Mg sheet. Int. J. Plast. 23 (1), 44–86.Marciniak, Z., Kuczynski, K., 1967. Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609–620.Mellor, P.B., Parmar, A., 1978. Plasticity analysis of sheet metal forming. In: Koistinen, D.P., Wang, N.-M. (Eds.), Mechanics of Sheet Metal Forming. Plenum

Press, New York, pp. 53–74.Miyauchi, K., 1984. A proposal of a planar simple shear test in sheet metals. Sci. Papers Inst. Phys. Chem. Res. 78 (3), 27–40.Miastkowski, J., Szczepinski, W., 1965. An experimental study of yield surfaces of prestrained brass. Int. J. Solids Struct. 1 (2), 189–194.Michno Jr., M.J., Findley, W.N., 1976. An historical perspective of yield surface investigations for metals. Int. J. Non-linear Mech. 11, 59–82.Mohr, D., Oswald, M., 2008. A new experimental technique for the multi-axial testing of advanced high strength steel sheets. Exp. Mech. 48, 65–77.Mohr, D., Dunand, M., Kim, K.H., 2010. Evaluation of associated and non-associated quadratic plasticity models for advanced high strength steel sheets

under multi-axial loading. Int. J. Plast. 26, 939–956.Moriya, T., Kuwabara, T., Kimura, S., Takahashi, S., 2010. Effect of anisotropic yield function on the predictive accuracy of surface deflection of automotive

outer panels. Steel Res. Int. 81 (9), 1384–1387.Sowerby, R., Duncan, J.L., 1971. Failure in sheet metal in biaxial tension. Int. J. Mech. Sci. 13 (3), 217–229.Spitzig, W.A., Richmond, O., 1984. The effect of pressure on the flow stress of metals. Acta Metall. 32 (3), 457–463.Stout, M.G., Hecker, S.S., Bourcier, R., 1983. An evaluation of anisotropic effective stress–strain criteria for the biaxial yield and flow of 2024 aluminum tubes.

Trans. ASME J. Eng. Mater. Technol. 105 (1983), 242–249.Sumita, T., Kuwabara, T., Hayashida, Y. 2011. Measurement of work hardening behavior of pure titanium sheet using a servo-controlled tube bulge testing

apparatus. In: The 14th International ESAFORM Conference on Material Forming, AIP Conference Proceedings, 1353, pp. 1423–1428.Tozawa, Y., 1978. Plastic deformation behavior under conditions of combined stress. In: Koistinen, D.P., Wang, N.-M. (Eds.), Mechanics of Sheet Metal

Forming. Plenum Press, New York, pp. 81–110.Vegter, H., van den Boogaard, A.H., 2006. A plane stress yield function for anisotropic sheet material by interpolation of biaxial stress states. Int. J. Plast. 22,

557–580.Verma, R.K., Kuwabara, T., Chung, K., Haldar, A., 2011. Experimental evaluation and constitutive modeling of non-proportional deformation for asymmetric

steels. Int. J. Plast. 27, 82–101.Von Mises, R., 1913. Mechanik der festen Korper un plastich deformablen Zustant. Göttingen Nachrichten, Math.-Phys. Klasse, pp. 582–592.Wagoner, R.H., Wang, N.-M., 1979. An experimental and analytical investigation of in-plane deformation of 2036-T4 aluminium sheet. Int. J. Mech. Sci. 21,

255–264.Wagoner, R.H., Laukonis, J.V., 1983. Plastic behavior of aluminum-killed steel following plane-strain deformation. Metall. Trans. 14A, 1487–1495.Williams, J.F., Svensson, N.L., 1970. Effect of tensile prestrain on the yield locus of 1100-F aluminium. J. Strain Anal. 5 (2), 128–139.Yanaga, D., Kuwabara, T., Uema, N., Asano, M., 2012. Material modeling of 6000 series aluminum alloy sheets with different density cube textures and effect

on the accuracy of finite element simulation. Int. J. Solids Struct. 49, 3488–3495.Yoon, J.W., Barlat, F., Dick, R.E., Chung, K., Kang, T.J., 2004. Plane stress yield function for aluminum alloy sheets—Part II: FE formulation and its

implementation. Int. J. Plast. 20, 495–522.Yoshida, K., Kuwabara, T., Narihara, K., Takahashi, S., 2005. Experimental verification of the path-independence of forming limit stresses. Int. J. Form.

Process. 8, 283–298.Yoshida, K., Kuwabara, T., 2007. Effect of strain hardening behavior on forming limit stresses of steel tube subjected to nonproportional loading paths. Int. J.

Plast. 23, 1260–1284.Yoshida, K., Kuwabara, T., Kuroda, M., 2007. Path-dependence of the forming limit stresses in a sheet metal. Int. J. Plast. 23 (3), 361–384.

multiaxial tube expansion test method for measurement of sheet metal deformation behavior under...

Documents