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Journal of Materials Processing Technology 213 (2013) 961970

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Journal ofMaterials Processing Technology

journal homepage: www.elsevier .com/ locate / jmatprotec

Numerical verification ofa biaxial tensile test method using a cruciform specimen

Yasuhiro Hanabusaa, Hideo Takizawab, Toshihiko Kuwabara c,

a DevelopmentDepartment, Universal Can Corporation, 1500Suganuma, Oyama-cho, Sunto-gun, Shizuoka 410-1392, Japanb Central Research Institute,Mitsubishi Materials Corporation, 1975-2 Shimoishitokami, Kitamoto-shi, Saitama 364-0022, Japanc Division of AdvancedMechanical Systems Engineering, Institute of Engineering, TokyoUniversity of Agriculture and Technology, 2-24-16 Naka-cho, Koganei-shi,

Tokyo 184-8588, Japan

a r t i c l e i n f o

Article history:

Received 6 October 2012Received in revised form

18 December 2012

Accepted 19 December 2012

Available online 28 December 2012

Keywords:

Sheet metal

Finite element method

Biaxial tension

Isotropy

Yield function

a b s t r a c t

A method ofevaluating stress measurement errors in biaxial tensile tests using a cruciform specimen is

proposed. The cruciform specimen is assumed to be fabricated from a section ofuniformly thick flatsheet

metal via laser or water-jet cutting and to have a number ofparallel slits cut into each ofthe four arms.

Using finite element analyses with the von Mises yield criterion, the optimum geometry ofthe cruciform

specimen and the optimum strain measurement position necessary to minimize the stress measurement

error are determined. Additionally, an experimental validation of the FEA is performed using a sheet

material that was experimentally confirmed to be nearly isotropic. The following conclusions are drawn:

(i) the thickness of the test material should be less than 0.08B (B: side length of the gauge area of the

cruciform specimen); (ii) the geometric parameters for the cruciform specimen should be N 7, L B,

ws 0.01B, and 0.0034 R/B 0.1 (N: number ofslits, L: length ofslits cut into the arms, ws: slit width,and R: corner radius at the junction of the arms to the gauge area); and (iii) the strain components in

the gauge area should be measured on the centerline of the specimen parallel to the maximum force

direction at a distance ofapproximately 0.35B from the center ofthe specimen. The stress measurement

error is estimated to be less than 2% when the optimum conditions above are satisfied.

2012 Elsevier B.V. All rights reserved.

1. Introduction

Highly accurate material constitutive equations are required in

order to perform accurate sheet metal forming simulations using

finite element analysis (FEA), and material testing under multiaxial

stresses is indispensable for establishing these equations.

One of the typical methods for biaxial stress testing of sheet

metals utilizes cruciform specimens, as reviewed by Kuwabara

(2007) and Hannon and Tiernan (2008). This test method has sev-

eral advantages, including the ability to measure the elasticplastic

deformation behaviorof sheetmaterials for an arbitrarystress ratio

and its immunity to the out-of-plane deformations that occur in

hydrostatic bulge testing.

However, unlike uniaxial tensile testing, the stress distributionin the gauge area of a cruciform specimen is not uniform, which

complicates the determination of the biaxial stress components.

Therefore, the cruciform specimen geometry and the method used

for measuring biaxial stress and strain components are the most

important factors for establishing a reliable and accurate biaxial

tensile test method for sheet metals.

Corresponding author.

E-mail addresses: [email protected] (Y. Hanabusa), [email protected]

(H. Takizawa), [email protected] (T. Kuwabara).

In this study, cruciform specimens are classified into types A

and B. TypeA has a reduced thickness area (henceforth, referred

to as gauge area) for measuring biaxial stress components, while

type B has uniform thickness over the whole specimen geometry.

Many researchers have investigated biaxial tensile test methods

using type A specimens. Pascoe and de Villiers (1967) proposed

a cruciform specimen with spherical recesses on both sides of

the central region for use in investigating the short-life fatigue of

mild and heat-treated steel types under biaxial loading. Shiratori

and Ikegami (1968) proposed a cruciform specimen consisting of

one cross-shaped sheet sample and eight plates for reinforcing the

four arms. This specimen was used to measure the yield loci of a

brass sheet that had been prestrained along various loading paths.

Hayhurst (1973) developed a biaxial tension creeprupture test-ing machine and a cruciform specimen with a uniformly thinned

central region and slots in the arms. Makinde and Ferron (1988)

measured the plane strain and equibiaxial work hardening behav-

ior of aluminum-1050A, (70/30) brass and austenitic stainlesssteel sheets using a novel and inexpensive biaxial device (jointed-

arm mechanism) developedby Ferron andMakinde(1988). Boehler

et al.(1994) developed a new screw-driven biaxial testing machine

that used a specialized off-axis testing device and cruciform spec-

imens with an optimized design to allow almost perfect off-axis

biaxial tensile testing of anisotropic sheet materials. Hjelm (1994)

successfully measured the yield locus for gray cast iron in the

0924-0136/$ seefrontmatter 2012 Elsevier B.V. All rights reserved.

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962 Y. Hanabusa et al. / Journal of Materials Processing Technology213 (2013) 961970

third quadrant (compressivecompressive) of stress space. Yuetal.

(2002) proposed an optimum specimen geometry for measuring

limit strains. Green et al. (2004) measured biaxial stress strain

curves for a commercial 1145 aluminum alloy sheet deformed up

to effectivestrains of approximately 0.15along sevendifferent pro-

portional strain paths. The specimen had a sandwich design with

the sample sheet bonded by adhesive between two face sheets

while leaving the central area exposed on both sides; the speci-

men geometry was the same as that developed by Makinde et al.

(1992).

Because the fabrication of typeA specimens requires a machin-

ingprocessor sandwichdesign to reducethe thickness of thegauge

area, it is difficult and expensive. Furthermore, the geometrical

constraint of the arms on the deformation of the thinned gauge

area makes it difficult to accurately determine the biaxial stress

components in the gauge area.

Type B specimens are easy to fabricate from flat sheet metals by

laser or water-jet cutting and are, therefore, less expensive. Sev-

eral authors used type B specimens with no slits cut into the arms.

Kreiig and Schindler (1986) determined yield loci of as-received

and prestrained (uniaxial and equibiaxial) sheet metal; the propor-

tional elastic limit was used for the definition of yielding. Mller

and Phlandt (1996) optimized the cruciform specimen geometry

using a FEA and photoelastic tests; the yield point was deter-mined through measurement of the temperature using an infrared

thermocouple. However, they pointed out the difficulty in defin-

ing the effective cross sectional area of the cruciform specimen.

Hallfeldt (2002) proposed to optimize an angle between specimen

arms depending on the degree of anisotropy of the material. All

the above-mentioned authors proposed cruciform specimens with

different geometries. However, whatever the geometry of the cru-

ciform specimen is, producing a uniform stress distribution in the

gauge area is difficult. This is because the four arms connecting to

the gauge area prohibit uniform deformation. As a result, it is dif-

ficult to accurately determine the biaxial stress components in the

gauge area.

Based on the above, from the viewpoints of fabrication cost and

stress uniformity in the gauge area, the use of type B cruciformspecimens with slits cut into the arms is considered most promis-

ing for accurate biaxial tensile testing of sheet metals. Kuwabara

et al. (1998) performed biaxial tensile tests of a cold-rolled, ultra-

low-carbon steel sheet using a cruciformspecimenwith seven slits

cutin each arm. It wasfound that themeasureddirections of plastic

strain rates were in good agreement with those of the local out-

ward vectors normal to the contours of plastic work constructed

in the principal stress space, and that Hills quadratic yield crite-

rion (Hill, 1948) overestimates the flow stresses in the vicinity of

equibiaxial tension. Kuwabara et al. (2000) successfully detected

the yield vertex and non-normality behavior of the plastic strain

rate for ultra-low-carbon steel sheet and aluminum alloy sheet

using the abrupt strain path change method proposed by Kuroda

and Tvergaard (1999). Kuwabara et al. (2002) performed biaxialtensile experiments of six kinds of steels with different r-values,

using thesame cruciformspecimen as developedby Kuwabaraet al.

(1998). It was found that the TaylorBishopHill (TBH) model with

the full constraints assumption is superior to that with the relaxed

constraints assumption in predicting the plastic deformation char-

acteristics of the test materials. Borsutzki et al. (2002) measured

a true stressstrain curve for equibiaxial tension of a DC04 steel

sheet using a cruciform specimen with six slits in each arm, up to

an equivalent plastic strain of 0.09 with the use of a laser exten-

someter. Naka et al. (2003) investigated the effect of temperature

on the yield locus of 5083 aluminum alloy sheet using a cruci-

form specimen with three slits in each arm. Geiger et al. (2005)

recommended cutting six slits into each arm based on FEA results.

Kulawinskiet al. (2011) applied thepartialunloading method to the

measurement of the yield locus of cast TRIP steel using a cruciform

specimen with three slits in each arm. However, even though all

the above-mentioned authors used typeB specimens with slits cut

into the arms, stress measurement errors and the optimum spec-

imen geometry were not clarified in those studies. Hoferlin et al.

(2000) found that theyield loci measured at a 0.2% von Mises equiv-

alent plastic strain for DC06 (cold-rolled, ultra-low-carbon steel)

and ZStE 220 BH (bake-hardening steel) were in good agreement

with the theoretical predictions based on the TBH model with full

constraints using experimentally determined crystallographic tex-

tures. They used a type B specimen that had a square metal sheet

gauge area with thin metal bars welded to the sides of the gauge

area; the geometry of the specimen was optimized using texture-

based anisotropic FEA. However, fabrication of such specimens is

time consuming and expensive, and the welded zones may impart

unfavorable effects on boththe uniformityof the stress distribution

and the mechanical properties of the gauge area.

There have been ongoing efforts to optimize the specimen

geometry by quantitatively evaluating stress and strain distribu-

tions in the gauge area using FEA. Makinde et al. (1992) optimized

the geometry of a cruciform specimen with a circular reduced

region and a type B cruciform specimen with seven slots cut into

each arm, using a statistical methodcoupled with a FEA. Demmerle

and Boehler (1993) performed a FEA for a type A specimen withslits cut into the arms to estimate the standard deviations of the

stresses andstrainsin the gauge area and to realize shape optimiza-

tion of biaxial cruciform specimens for isotropic elastic materials.

Lin and Ding (1995) performed a FEA for a typeA specimen with

slits cut into the arms to determine an optimum value of the effec-

tive cross sectional area of the gauge area, even though the FEA

was performed only for the case of equibiaxial tension. Ikeda and

Kuwabara (2002) proposed a planestrain tension specimen with

slits cut into the arms and determined the optimum combination

of slit number and slit tip hole diameter to minimize the stress

measurement error. However, none of these studies discussed the

optimum strain measurement position for minimizing stress mea-

surement errors; nor did they quantitatively consider the effects

of the geometrical parameters of the cruciform specimens on thestress measurement errors for various stress ratios.

The objective of this study is to clarify the optimum strain

measurement position and geometrical parameters for the type B

specimen geometry proposed by Kuwabara et al. (1998) for min-

imizing the stress measurement error, on the basis of FEA using

the von Mises yield criterion. Additionally, an experimental vali-

dation of the FEA is performed using a sheet material that was

experimentally confirmed to be nearly isotropic.

It should be noted that the biaxial tensile test method using a

cruciform specimen proposed in this study has proven to be useful

for accurately detecting and modeling the deformation behavior of

sheet metals under biaxial tension, and consequently improves the

predictive accuracy of the FEA for springback in stretch-bending

(Kuwabara et al., 2004), surface deflection of an automotive bodypanel (Moriya et al., 2010), hole expansion of high strength steel

sheet with a tensile strength of 590MPa (Hashimoto et al., 2010)

and 780 MPa (Kuwabara et al., 2011), and hydraulic bulge forming

of 6000 series aluminum alloy sheet (Yanaga et al., 2012).

2. Evaluationmethod for stressmeasurement error

In this section, we describe our method for evaluating the

accuracy of stress measurements in a biaxial tensile test using a

cruciform specimen. FEA is used to determine the optimum geo-

metrical parameters for the cruciform specimen and the optimum

position for strain measurement to minimize the stress measure-

ment error.

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Y. Hanabusa et al. / Journal of MaterialsProcessing Technology213 (2013) 961970 963

Fig. 1. Geometry of cruciform specimen.

Fig. 1 shows the cruciform specimen geometry investigated in

this study. The geometry of the specimen was originally proposedby Kuwabara et al. (1998), and was fabricated from flat sheet metal

via laser cutting. This specimen consists of a square gauge area

(the shaded area in Fig. 1), for which biaxial stress components are

determined, and four arms that extend from the gauge area. Paral-

lel slits are cut into each arm to ensure that the stress distribution

is kept as uniform as possible in the gauge area, thus minimizing

the stress measurement error to the greatest extent possible. The

extremity of each arm is held in the grip of a biaxial tensile testing

machine.

In Fig. 1, B is the width of the arms, and BSx and BSy are thedistances between the slit tipsalong thex-andy-axes(placedalong

the arm midlines), respectively. It is assumed in this study that

BSx = BSy = B. L is theslit length,ws is theslit width, Nis the number

of slits in each arm, and R is the corner radius at the junction of thearms to thegauge area. Theslit tips aresemicircular with a radiusof

ws/2. The initial thickness of the specimen is t0, which is the sameas the as-received sheet sample.

In contrastto conventionaluniaxial tensionspecimens, it is diffi-

cultto achieve a uniform distribution of biaxial stresses in thegauge

area of a cruciform specimen. Therefore, the objective of this study

is to determine the optimum strain measurement position and the

parameters, L, ws, Nand R, which define the specimen geometry,so that the stress measurement error is minimized. Additionally,

the effects of the initial thickness of the specimen t0 and the work-hardening exponent n on the stress measurement error are also

examined.

The present approach to evaluating the stress measurement

error is explained using the conceptual illustration shown in Fig. 2.It is assumed that the biaxial strain components in the gauge area

are measured ata single point inthe gauge areausing a sensorsuch

as a strain gauge.

In orderto determinean accuratestressstrainrelationshipfor a

sheet material subjected to biaxial tension, it is crucial to measure

the local stress components L (Lx , Ly) as accurately as possi-

ble, at the position where the local strain components (x, y) aremeasured.

However, the measurable stress components in the biaxial ten-

sile test are the averages G (Gx , Gy ), determined by dividing

the biaxial tensile forces, Fx and Fy, by the respective current cross-sectional areasof the gauge area.Consequently, themost important

requirement for establishing a highly accurate biaxial stress test

methodistomakeG

approach L

ascloselyaspossible.Itshouldbe

Fig. 2. Schematic illustration of average stress G and local stressL.

noted thatL cannot be measured directly in an actual experiment,but it is possible to estimate the value ofL using FEA.

In the followingsection, a methodfor evaluating thestress mea-

surement errors using FEA is proposed. This method allows usersto determine the optimum specimen geometry and the optimum

strain measurement position necessary for minimizing the stress

measurement error.

2.1. Evaluation of stress measurement error

In thissection,we define the stress measurement errores, which

is an indicator to show how close G is toL. The relation betweenL and (x, y) at an integration point in the FEA model follows thematerial constitutive equations used in the FEA. G is calculatedusing Eq. (1):

Gx =Fx exp(x)

Bt0(1a)

Gy =Fy exp(y)

Bt0(1b)

in which a uniform deformation in the gauge area and a constant

volume condition are assumed.

Then, the stress measurement error es is defined as follows:

es |G L|

|L|=

(G

ij L

ij)(G

ij L

ij)

LklLkl

(2)

whereit isassumedthatG12 = G23 =

G31 =

G33 = 0and

L23 =

L31 =

L33 = 0 (for the case ofplanestresselements).The positionat which

es becomes a minimum gives the optimum strain measurement

position.

2.2. Finite element model and analysis conditions

The cruciform specimen geometry shown in Fig.1 was usedasa

model specimen forFEA. The clamping area wasmodeled by a rigid

connection of nodes along theedge of thearm. Thelateral displace-

ments of the nodes were fixed, and the combined tensile forces in

thex- andy-directions were applied to the respective connections

(see Fig. 5).

MSC.Marc2008r1 was used for the analysis. Because of the sym-

metryof thespecimen,onlyone quarter of the samplewas modeled

and 2D analyses wereperformedusing plane stresselements, while

3D analyses were also performed using solid elements when eval-

uating the effect of specimen thickness on es, in which one half

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Table 1

Analysis conditions for FEA (underline: standard conditions).

Stress ratio: sx : sy 1:0, 4:1, 2:1, 4:3, 1:1

Thickness: t0 (mm) 0.6, 1.2, 2.4, 3.6, 4.8

Slit length: L (m m) 15, 30, 45, infin it e len gt h ( un con st raint alo ng

arm edges in transversal direction)

Numberof slits:N 3, 5, 7, 9

Slit width: ws (mm) 0.2, 0.3, 0.5

Corner radius: R (mm) 0.1, 0.5, 1.0, 3.0

Table 2

Work hardening characteristics of material model assumed in FEA (underline:

standard condition).

n-valuea 0 a ca (MPa)

fs-0 0.209 0.0041 522

fs-1 0.2 0.0038 505

fs-2 0.3 0.0073 723

fs-3 0.4 0.0110 1004

a Swift type flow curve: = c(0 + p)n

.

volume in the thickness direction of the specimen was modeled. It

was assumed that BSx = BSy = B = 30 mm.Table 1 shows the analysis conditions for the FEA. The nominal

stress ratios chosen were sx : sy = 1 : 0,4 : 1,2 : 1,4 : 3 and 1 :1. The effects of the geometrical parameters, t0, L, ws, N and R,on es were investigated in detail. The underlined values in Table 1indicate the standard conditions for the FEA.

Regarding the material model used in the FEA, isotropy is

assumed in both elasticity and plasticity; Youngs modulus was

200 GPa, Poissons ratio was 0.33, and the Von Mises yield crite-

rion was used. Table 2 shows the work hardening characteristics

of the material model assumed in the FEA. The effect of the work-

hardening exponent n on es was also evaluated. The data, fs-0, forthecold-rolled ultralowcarbonsteel sheettestedby Kuwabaraetal.

(1998) was used as a standard condition. The parameters for fs-1

through-3 were determined so that they have thesame initial yield

stress and the flow stress at p = 0.01 as those for fs-0.

3. Results and discussion

3.1. Optimum strain measurement position

Fig. 3 shows the contour diagrams for es calculated for thestandardconditionsat an equivalent plastic strain of approximately

Fig. 3. Distribution of stress measurement error es in gauge area.

Fig. 4. Distributionof stresscomponents in gauge area. sx : sy: nominal stressratio

applied to specimen.

0.01. As the nominal stress ratio sx : sy changes from uniaxialtension (sx : sy = 1 : 0) to equibiaxial tension (sx : sy = 1 : 1), esbecomes more uniform and smaller in the gauge area.

Fig. 4 shows the stress distributions in the gauge area. For sx :sy = 1 : 1 , x is mostly uniform over a wide area from the centralareato the vicinityof the slit tips. For sx : sy = 4 : 3,the distributionof y in the vicinity of the slit tips shows nonuniformity over a

somewhat wider area than that for sx : sy = 1 : 1, but otherwise,the uniformity of stress distribution is almost identical.

Comparing the stress distributions for sx : sy = 2 : 1,4 :1 and 1 : 0, it is noted that the degree of stress nonuniformity

for y becomes more significant than that for x as the stress

ratio approaches uniaxial tension, sx : sy = 1 : 0. This is because

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150 30 45 60 75 90-45

0

45

90

135

x

y

p

0= 0.01

Exp.

von Mises

Hill'48Yld2000-2d

(M= 5)

Directionofplasticstrainrate

/

Loading direction /

Fig. 14. Comparison of the directions of measured plastic strain rates with those

ofthelocal outward vectors normal to theyield loci calculatedusing selected yield

functions.

analysis system (GOM, ARAMIS). True stress components (x, y)

were determined by dividing (Fx, Fy) by the current cross-sectional

area of the gauge section, which was determined from the mea-

sured values of the plastic strain components px ,

py with an

assumption of constant volume. xy was assumed to be zero, asx and y were measured on the centerlines of the specimen. Forx : y = 1 : 0 and 0:1, standard uniaxial tensile specimens, JIS 13B-type (JIS Z2241), were used. The equivalent plastic strain rate

was (56)104 s1. Details of the biaxial testing apparatus and

test method are given in Kuwabara et al. (1998, 2000).

4.3. Results of biaxial tensile tests

Fig. 13 shows the measured stress points that form the con-

tours of plastic work for p0 = 0.01. All of the stress values of the

respective stress points constituting the work contour are normal-

ized by 0 corresponding to p0 = 0.01. Also depicted in the figures

are the theoretical yield loci based on the Von Mises (1913), Hills

quadratic (Hill, 1948) and the Yld2000-2d yield functions with an

exponent ofM= 5 (Barlat et al., 2003). The unknown parame-

ters of the Hills quadratic yield function were determined using

r0, r45, r90 and 0/0, while those of the Yld2000-2d yield func-tion were determined using r0, r45, r90 and rb and 0/0, 45/0,90/0 and b/0, where r and are ther-value and tensile flow

stress measured at an angle from the RD,respectively, and whererb and b are the ratio of the plastic strain increments, d

py/d

px ,

and the flow stress at equibiaxial tension, x : y = 1 : 1, respec-tively. The values ofr0, r45 and r90 used were the same as those

in Table 3, for both the Hill 48 and Yld2000-2d yield functions.

The values of45/0, 45/0, 90/0, b/0 and rb used to deter-mine the Yld2000-2d yield function correspond to those defining

the work contour shown in the figure. The reason why the expo-

nent M= 5 was chosen for the Yld2000-2d yield function was thatit gave a smaller standard deviation from the work contour than

M= 4 and 6; see Yanaga et al. (2012) for the standard deviationcalculation method. It is clear that the Yld2000-2d yield function

with M= 5 agrees better with the work contour than other yieldfunctions.

In order to validate the normality flow rule for the yield func-

tions used in Fig. 13, the directions of plastic strain rates, , were

Fig. 15. Contour diagrams for total strains x and y measured for x : y = 4 : 3 a t x = 0.02 (a), compared with those calculatedusing FEA (b).

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Y. Hanabusa et al. / Journal of MaterialsProcessing Technology213 (2013) 961970 969

Table 4

Material parameters for theYld2000-2d (M= 5) yield function for p

0= 0.01.

1 2 3 4 5 6 7 8

0.9572 0.9905 0.9781 0.9686 1.014 0.9715 0.9791 1.0316

0.00 0.01 0.020

200

400

600

800

x:

y= 1 : 0

x:

y= 4 : 3

Experimental

FEA

y

x

Truestress

/MPa

True plastic strainp

Fig. 16. Biaxial stressstrain curves measured for x : y = 4 : 3, compared with

those calculated using FEA.

measured for all linear stress paths and compared with those of

the outward vectors normal to the theoretical yield loci at corre-

sponding stress points. The results are shown in Fig. 14, where isthe loading angle of a stress path from the x-axis in the principal

stress space, and both and are defined as zero along thex-axisand positive in the anti-clockwise direction. The Yld2000-2d yield

function withM= 5 again provides the closest agreement with themeasurement.

Consequently, we conclude that the Yld2000-2d yield function

with M= 5 is an appropriate material model for the test materialunder linear stress paths. Table 4 shows the material parameters

1 8 for the Yld2000-2d yield function with M= 5.

4.4. Validation of the FEA

Fig. 15 shows the contour diagrams for total strains x and ymeasured forx : y = 4 : 3 a t x = 0.02, compared with those cal-

culated using FEA. The FEA was performed using the Yld2000-2d

yield function with M= 5, as determined in Section 4.3. The mea-sured and calculated results for both x and y were found to be ingood agreement with each other.

Fig. 16 shows the biaxial stressstrain curves measured forx : y = 4 : 3,compared withthose based on the FEA for the samestress ratio. For the FEA, the calculation procedure used for deter-

miningtruestress andtrueplastic straincomponents wasthe same

as that used in the experiment. The FEA result agreed well with the

measurement.From the results shown in Figs. 15 and 16, it can be concluded

that the validity of the FEA as described in Sections 2 and 3 has

been experimentally verified. This result is consistent with the

conclusion obtained by Hanabusa et al. (2010) and indicates that

the optimal strain measurement position for an isotropic mate-

rial, as clarified in this study, also holds true for anisotropic sheet

materials.

5. Conclusions

This paper investigated a method for determining the opti-

mum strain measurement position for minimizing the stress

measurement error es in a biaxial tensile test method usingthe cru-

ciform specimen developed by Kuwabara et al. (1998). The effects

ofthe geometrical parameters of thecruciform specimenon es were

also examined in detail. es is estimated to be less than 2% when theconditions (1), (2) and (3) are satisfied:

1) The thickness t0 of the cruciform specimen is less than 0.08B,where B is the side length of the gauge area.

2) N 7, L B and ws 0.01B, where N is the number of slits ineach arm, L is the slit length, and ws is the slit width of the

cruciform specimen. The effect of the corner radius R on es isnegligibly small when it is in the range of 0.0034 R/B 0.1.

3) The biaxial strain components are measured at the position(s)

onthe centerline of the specimen,parallel to the maximum force

direction, at a distance of approximately 0.35B from the center

of the specimen.

The validity of the FEA was experimentally verified by compar-

ing the FEA results with those measured from the biaxial tensile

tests of a sheet material that was experimentally confirmed to be

nearly isotropic.

Acknowledgments

The authors would like to express their gratitude for the indis-pensable advice we received from the members of the Committee

on Standardization of Biaxial Stress Test Method, which has been

supporting the standardization project since 2008 under the aus-

pices of the Osaka Science and Technology Center on commission

from the New Energy and Industrial Technology Development

Organization (NEDO).

Appendix A.

For the cruciform specimen investigated in this study, the arms

are subjected to uniaxial tension, which means that the test is

complete when the nominal stress in the arms reaches the ten-

sile strength of the material. Accordingly, the maximum equivalent

plastic strain pmax applicable to the gauge area of the cruciformspecimen can be estimated using the considere condition for a

maximum load on a strip in tension (Marciniak et al., 2002). pmax

depends primarily on the stress ratio, the work hardening expo-

nent,n, andthe anisotropy of the test material,and theslit widthwscut into each arm of the cruciform specimen. This Appendix shows

theeffectsofnandws onpmax.Itshouldbenotedherethatthesecal-

culation results should be viewed only as a reference, because they

are numerical analysis solutions based on the simple mechanics of

plasticity assuming an isotropic material.

Fig. A.1 shows the effect of the number of slits, N, on pmax

for a cruciform specimen with B = 30 mm and ws = 0.2,0.3 and0.5mm subjected to a true stress ratio ofx : y = 2 : 1. For thestandard conditions (N= 7 and ws = 0.2 mm),

pmax = 3.2%, but

decreases to 2.9%whenN= 9. Whenws = 0.5 mm, pmax decreasesto 1.5% for N= 9. This is because the effective cross-sectional area

Fig.A.1. Effect of number of slitsNandslit widthws on maximum equivalentplastic

strain

p

max applicable to gauge area of cruciform specimen.

7/27/2019 1-s2.0-S0924013612003780-main

10/10


0.005 0.010 0.015 0.0200.00

0.02

0.04

0.06

0.08

0.10

0.12

p max

s

n =0.25

n =0.10

x:

y=2:1

n =0.20

n =0.30

w /B

(a)

0.005 0.010 0.015 0.0200.00

0.02

0.04

0.06

0.08

0.10

0.12

p m

ax

s

n =0.30

n =0.1

x:

y=1:1

n =0.20

n =0.25

w /B

(b)

Fig.A.2. Effectof slit widthws and workhardeningexponent non maximum equiv-

alentplasticstrainpmax applicableto gaugearea of cruciformspecimen. Thenumber

ofslits is 7.

of the arms decreases with increasing N, although the stress distri-

bution in the gauge area becomes more uniform.

Fig. A.2 shows the effects ofws and n on pmax with N= 7 for x :

y = 2 : 1 and 1:1. pmax decreases with increasing ws, because the

effective cross-sectional area of the arms decreases, which results

ina decrease inthe maximumforcetransmitted tothe gaugeareaof

the cruciform specimen. pmax increases withn, because a material

with a larger n-value is subject to a higher stress increase rate as

the plastic deformation of the arms increases, whichin turn causes

an increase in the tensile stress acting on the gauge area.

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