Experimental and Numerical Analysis of an In-Plane ShearSpecimen Designed for Ductile Fracture Studies

F. Gao & L. Gui & Z. Fan

Received: 28 November 2009 /Accepted: 28 June 2010# Society for Experimental Mechanics 2010

Abstract An in-plane shear specimen made of dual phasesteel designed for ductile fracture studies is presented andthen analyzed experimentally and numerically. In theexperiment, digital image correlation (DIC) technique isutilized to measure the deformation of the specimen. Basedon the implicit nonlinear FE solver Abaqus/Standard,numerical analysis of the specimen is performed by usingplane stress and solid elements respectively. The elongationof the specimen’s gauge length and the shear straindistribution within the shear zone are compared betweenthe experimental and numerical results and a general goodagreement is obtained. Thereafter, based on calculatedresults, the stress state of the shear zone is investigated indetail. It is shown that the shear stress is dominant withinthe shear zone despite of the emergence of normal stresses.The deformation is concentrated in the shear zone, wherethe incipient fracture is most likely to occur. The stresstriaxiality and the Lode parameter at the fracture initiationare found to be maintained at a relatively low level, whichimplies that the stress state achieved by the specimen isclose to pure shear. The present study demonstrates that theproposed in-plane shear specimen is suitable for investiga-tion of the fracture behavior of high strength materialsunder shear stress states.

Keywords In-plane shear specimen . Fracture . Dual phasesteel . Digital image correlation . Numerical simulation

Introduction

Advanced High Strength Steels (AHSS) consisting of DualPhase (DP) steel, TRansformation Induced Plasticity(TRIP) steel and Complex Phase (CP) steel have a plastichardening process relying on phase transformation. Thebiggest advantage of AHSS lies in its high strength, whichwould likely reduce automotive mass if it were applied toauto-vehicles [1]. However, compared with conventionaldeep drawing steels, the ductility is poor, which can beexhibited from the short elongations at fracture. Frequentfailures due to fracture have been observed during theforming processes. Therefore, the fracture investigations ofAHSS have become one of the hottest topics for currentautomotive lightweight researchers [2].

Shear tests are of great importance since they are widelyused to investigate the shear modulus and shear strength ofmaterials. The traditional shear test is performed by rotating acylindrical specimen through a torsion testing machine,although this procedure cannot be applied to sheet metals.The simple shear test method seems to be an effectivetechnique for evaluating the mechanical properties of planarspecimens [3, 4]. In this test the specimens have a simplerectangular shape and are easy to prepare. This test requires aspecial device which mainly consists of two rigid parts withone piece remaining fixed and the other piece moving in thevertical direction. During testing, the planar specimen isgripped between the two parts and one side of the specimenremains stable while the other side moves up or down.However, premature fracture may occur by cracking along thegripping lines subjected to tension [4]. Iosipescu [5]presented a new method to achieve the pure shear state byimposing asymmetrical four-point bending to a notchedplanar specimen. This approach has been widely used to testcomposite materials [6, 7]. However, the uniformity of the

F. Gao : L. Gui (*) : Z. FanState Key Laboratory of Automotive Safety and Energy,Department of Automotive Engineering, Tsinghua University,Beijing 100084, Chinae-mail: gui@tsinghua.edu.cn

Experimental MechanicsDOI 10.1007/s11340-010-9385-8

shear stress between the notch tips was found to be highlydependent on the elastic properties of the orthotropicmaterial [8].

Bao andWierzbicki [9] displayed a shear specimen with acomplex shape during fracture investigations of an alumi-num alloy. The shear zone of the specimen is concentrated inthe butterfly section with a reduced thickness used to create alow stress triaxiality state. The disadvantage of the specimenis that the reduced thickness is difficult to machine for thinsheet metals and will possibly introduce damage or microcracks on the surface of the shear zone, which may bedetrimental for incipient fracture judgment.

Tarigopula et al [10] designed a new shear specimengeometry based on extensive FE optimizations and demon-strated the use of the specimen to identify strain hardeningparameters for DP800 steel at large strains. As we know, astress state can be characterized by the stress triaxiality andthe Lode parameter. The stress triaxiality is defined as theratio of the mean stress and equivalent stress, and has beenfound to be an imperative factor influencing the growth ofmicro voids which lead to macro cracks [11–13]. The Lodeparameter is defined by the third stress invariant, whichmeasures the effects of deviatonic stresses. The relationfunction between the stress triaxiality and the Lode parameterunder plane stress states has been derived that it’s a thirdorder parabola [14]. In the in-plane pure shear state, both thestress triaxiality and the Lode parameter equal zero. However,this pure shear state is difficult to maintain in shear tests.From the failure parameter data displayed by Tarigopula et al[10], it was noticed that the stress triaxiality increases to 0.23,which means the stress state during the later stage ofdeformation gradually becomes far from pure shear.

Based on the work of Bao [9, 12] and Tarigopula et al [10],the present study shows another in-plane shear specimen witha different shear zone which is used to obtain relatively lowstress triaxiality and Lode parameter without reduction in

thickness. First, the specimen made of DP800 is investigatedexperimentally by a digital image correlation (DIC) techniquewhich is convenient to measure the displacement and strainfields. Second, by means of the implicit FE solver Abaqus/Standard, numerical analysis of the specimen is carried out toaccurately investigate the shear stress distribution and allin-plane stress components’ development. In order tostudy the shear specimen separately, the strain hardeningparameters for FE modeling is obtained from uniaxialtensile tests.

Experiments

Uniaxial Tensile Tests

DP800 supplied by Swedish Steel Works is considered in thisstudy, which is identical to the material used by Tarigopula etal [10]. The cold rolled sheet metal possesses a nominalthickness of 2.0 mm. According to GB/T228-2002 standard,the uniaxial tensile specimens are designed and cut from aDP800 sheet with the longitudinal direction along the sheet’srolling direction. The specimens only in the rolling directionare considered because Tarigopula et al [10] has proven thatthis kind of material exhibits weak plastic anisotropicbehavior. A total of two groups of the specimens areprepared and each group contains three duplicate samples,which are denoted as ‘H11, H12, H13’ and ‘H14, H15,H16’respectively. One group’s deformation is measured withthe extensometer and that of the other group with DICtechnique so that we are able to evaluate the accuracy of theDIC method. The DIC technique will be used to measuredeformation of the following in-plane shear specimen andgood accuracy of the method will guarantee the validity ofthe test data.

The uniaxial tensile test is conducted on a SHIMADZUtesting machine with a load capacity of 50 kN. In tests of

(a) (b)

Fig. 1 Experimental set-up for the first group specimen tests: (a)testing machine and the specimen; (b) extensometer and the specimen

Fig. 2 Experimental set-up for the second group specimen tests

Exp Mech

the first group specimens (Number: H11, H12, H13),Epsilon 3542-INS extensometer is employed to measurethe elongation of the 50 mm gauge length of the specimen,as shown in Fig. 1. The specimen is stretched underdisplacement control with an elongation velocity of 1 mm/min, corresponding to a strain rate of 0.0003/s. The testingmachine documents the correlation curves between theapplied force and elongation, which will be used todetermine the strain hardening parameters for DP800.Elastic parameters such as Young’s modulus, E, andPoisson’s ratio, ν, are precisely obtained by the strainrosette which is attached to the surface of the gauge lengthbeforehand.

In tests of the second group specimens (Number: H14,H15, H16), the DIC technique has been utilized to measurethe displacement and strain variations of the specimen. Asshown in Fig. 2, two digital cameras (DC) are arranged,which are identified as DC1 and DC2 respectively. DC1 is

placed normal to one surface of the specimen and acquiresan image of 1376×1035 pixels with 256 grey levels, whichincludes the deformation of the gauge length 50.0 mm. Thefunction of DC1 is identical to the extensometer in Fig. 1(b). DC2 is placed normal to the other surface andemployed to capture the images of a local area, so that wegain the longitudinal strain variations accurately. The twodigital cameras start working simultaneously and acquireone image every 2 s.

Shear Tests

Based on numerous FE calculations, the final geometry ofthe in-plane shear specimen for DP800 is obtained, asshown in Fig. 3. The shear zone concentrates on the centerof the specimen and utilizes the butterfly shape presentedby Bao and Wierzbicki [9]. Compared with the design byBao [9], the biggest difference is that the shear zone isunsymmetrical with respect to the x1-axis. As seen fromFig. 3(b), two concave sides of the shear zone move

(a)

(b)

O A

x2

x1

Fig. 3 Geometry of the in-planeshear specimen: (a) geometricaldimensions; (b) magnification ofthe shear zone

9955 ..33mmmm

Fig. 4 Three in-plane shear specimens Fig. 5 Experimental set-up for shear tests

Exp Mech

1.0 mm along the x2-axis up and down respectively. Theextreme point ‘A’ of the concave side has a horizontaldistance of 2.5 mm with the center point ‘O’ of the shearzone. The above mentioned design results in the shear zonehaving a nominal length of 5.0 mm along the x1-axis of thespecimen. The shear zone always undergoes a rotationduring the loading process, which will increase the tensilestress components. However, the tensile stresses are notneeded. So the two offset concave sides are introduced inpromise of guaranteeing the shear stress and meanwhilerepressing the tensile stresses.

Following the geometry shown in Fig. 3, three duplicatesamples are cut from DP800 sheet with the x1-axis alongthe rolling direction and designated as ‘H31, H32, H33’respectively, as illustrated by Fig. 4. A gauge length of95.3 mm has been marked on the surface of the samples.

Shear tests are carried out on an electronic universaltesting machine WDW with a maximum load of 100 kN.One end of the specimen is fixed and the other end isstretched at a speed of 0.5 mm/min. The DIC technique hasbeen utilized to measure the displacement and strain field ofthe specimen. As displayed in Fig. 5, two digital cameras

(DC) are arranged, which are identified as DC1 and DC2respectively. DC1 is placed normal to one surface of thespecimen and acquires an image size of 114 mm×86 mm,which contains the deformation information of the gaugelength of 95.3 mm. The function of DC1 is identical to alarge range scale extensometer, and thus the elongation ofthe gauge length can be obtained. The image obtained byDC1 is digitized into a sample of 1376×1035 pixels with256 grey levels and then stored into a computer for furtherprocessing. DC2 is placed normal to the other surface of thespecimen and employed to capture the images of the shearzone, which are helpful in accurately studying the strainfield of the shear zone. Like DC1, DC2 also digitizes thecaptured images into samples of 1376×1035 pixels with256 grey levels and stores them into a computer. The twodigital cameras start working simultaneously that is,acquiring one image every 2 s.

Digital Image Correlation

DIC is a non-contact optical technique used to measure thedeformation field of an in-plane object. It obtains themeasurement data by matching the grey intensity distribu-tion of two sequential acquired images taken ‘before’ and‘after’ deformation, which are considered as the referenceimage and deformed image respectively. The light intensityat the point (x1, x2) in the reference image can be expressedby the grey matrix, G(x1, x2), over a selected subset. Thepoint (x1, x2) moves to a new location after deformation andis referred as the point ( x1

’, x2’ ) in the deformed image, the

0 1 2 3 40

5

10

15

20

H11 fd H12 fd H13 fd H14 fd H15 fd H16 fd

Elongation d/mm

Loa

d f/

KN

Fig. 6 Applied force versus elongation for the six uniaxial tensilesamples

0 1 2 30.00

0.05

0.10strain from H13strain from H14 by DICstrain from H15 by DICstrain from H16 by DIC

Lon

gitu

dina

l tru

e st

rain

Elongation d/mm

Fig. 7 Comparison of the longitudinal true strain derived from the testdate by extensometer and DIC

0.0 0.1 0.2 0.30

300

600

900

1200

Tru

e st

ress

/MP

a

True strain

H13 test fitted curve

Fig. 8 Experimental and fitted curves of the true stress versus truestrain for DP800

Table 1 Mechanical property parameters for DP800 from uniaxialtensile tests

E /GPa σ0.2 /MPa ν K /MPa n

210 556 0.3 1185.4 0.1179

Exp Mech

light intensity of which can be described as G’ ( x1’, x2

’ ).By searching the position of G’ ( x1

’, x2’ ) that mostly

resembles the original intensity G(x1, x2), the in-planedeformation measurements can be detected. The correlationcoefficient S is formulated to describe the similarity degreebetween G(x1, x2) and G’ ( x1

’, x2’ ) as follows [15, 16],

S ¼ 1�P

G x1; x2ð Þ»G0x01; x

02

� �� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

G x1; x2ð Þð Þ2»P G0 x01; x

02

� �� �2q ð1Þ

If the motion of the object relative to the camera isparallel to the image plane, then the coordinates (x1, x2) and( x1

’, x2’ ) are related by

x01;¼ x1 þ uþ @u

@x1Δx1 þ @u

@x2Δx2

x02 ¼ x2 þ vþ @v

@x1Δx1 þ @v

@x2Δx2

ð2Þ

where u and v are the displacements for the subset centersin the x1 and x2 directions respectively; @u @x1= ; @u @x2= ;

@v @x1= ; @v @x2= are the first-order displacement gradients ofthe reference subset; Δ x1, Δ x2 are the distances from point(x1 , x2 ) to the subset cente r. The u , v and@u @x1= ; @u @x2= ; @v @x1= ; @v @x2= are detected iteratively

from the position by minimizing S. Thereafter, thedeformation gradient tensor F can be determined by thefollowing definitions [17],

F11 ¼ @u@x1

þ 1; F12 ¼ @u@x2

F21 ¼ @v@x1

; F22 ¼ @v@x2

þ 1

ð3ÞDifferent strain definitions may be used to measure the

deformation. Here we make use of logarithmic strains,whose matrix εis expressed by the following formulationsignoring the deformation in thickness

" ¼ ln l1ð Þn1nT1 þ ln l2ð Þn2nT2 ð4Þ

where li and ni (i=1, 2) are the eigenvalue and thecorresponding unit eigenvector of the right Cauchy-Greendeformation tensor C, namely FTF[18].

Experimental Results

Uniaxial Tensile Tests

Resulting from the measurement range limitation of theextensometer, only a 3 mm elongation is acquired in the

(a) t=90s (b) t=260s (c) t=346s close to fracture

Fig. 9 Sequential imagescaptured by DC1 for thein-plane shear sample ‘H32’

(a) t=90s (b) t=260s (c) t=346s close to fracture

Fig. 10 Sequential imagescaptured by DC2 for thein-plane shear sample ‘H32’

Exp Mech

uniaxial tensile tests. In addition, by processing the imagescaptured by two cameras, we are able to obtain theelongation of the gauge length and the strain variationsduring loading. Then the test data can be compared to eachother. First, the curves between the applied force and theelongation of the 50 mm gauge length for the six samplesmeasured by two methods are compared and plottedtogether in Fig. 6. We can see a good agreement isobtained. Second, the true strains along the longitudinaldirection derived from the test data by the extensometer andmeasured by DIC are also compared, as shown in Fig. 7.It’s found they are consistent fairly well. From the twopoints discussed above, we may conclude that the used DICmethod is accurate and can provide an effective tool toinvestigate the deformation of the in-plane shear specimenin the following section.

Based on the results in Fig. 6, the relationship of truestress versus true strain characterizing the strain hardeningfor DP800 can be determined according to their definitions.A power function σ=Kεn has been adopted to fit the curveof true stress versus true strain, where K is the hardeningcoefficient and n denotes hardening exponent. Fig. 8depicts the two curves of true stress versus true strain fromthe tests and fitted results. There is no noteworthy yieldpoint observed and the yield stress is determined at a valueof 556 MPa corresponding to 0.2% plastic strain. Table 1presents the tested elastic parameters and fitted plastichardening parameters for DP800.

Shear Tests

A total of about 173 images were captured for each in-paneshear specimen. Using the sample ‘H32’ as an example,Fig. 9 describes the deformation evolution of the specimenin form of sequential images captured by DC1 andcorrespondingly Fig. 10 shows the deformation process ofthe shear zone by DC2. From the Figs. 9 and 10, it can beobserved that the shear zone comes through a large rotationbefore the occurrence of incipient fracture. Referring to thein-plane shear specimen designed by Bao [9] and Tarigopulaet al [10], the rotation of the shear zone is also detected andseems unavoidable in this kind of shear specimen. However,it should be recognized that the rotation would induce thestress components of the shear zone changing and increasingrotation would lead to larger normal stress componentswhich may be detrimental for the state dominated by shearstress. A detailed analysis on the stress distribution of theshear zone is studied accurately by using numericalcalculations. Furthermore, no thinning phenomenon inducedby plastic necking is observed, which implies the specimenmaintains a plane stress state during the deformation.

In order to obtain the relationship between the appliedforce and the elongation of the gauge length 95.3 mm, theDIC algorithm is employed to process the images acquiredby DC1. The curves obtained for the three samples areshown in Fig. 11. It can be seen that the curves transitsmoothly without any apparent drop before fracture, andthe elongation corresponding to the incipient fracture isapproximately 3.86 mm (Figs. 12 and 13).

By processing the images acquired by DC2, the shearstrain distribution within the shear zone can be determined,which are displayed in Fig. 14. We can see that the shearstrain distribution band goes through a finite rotation. Itagrees with what we observed in Fig. 10.

Numerical Simulation

The implicit solver of non-linear finite element code Abaqus/Standard is employed to simulate the quasi-static test of the in-plane shear specimen. The material constitutive equations arecompiled in Table 2. It takes the rate of deformation D andJaumann stress rate srJ as the energy conjugate measures.

0 1 2 3 4 50

2

4

6

8

H31 fd H32 fd H33 fd

Elongation d /mm

Load

f/K

Nfracture

Fig. 11 Relation curves between the applied force and the elongationof the gauge length for three in-plane shear samples

x1

x2

Fig. 12 The solid FE model forthe in-plane shear specimen

Exp Mech

Generally the rate of deformation D is decomposed into theelastic part De and the plastic Dp. Then the srJ and De areconnected with a four-order tensor Ce, which are determinedaccording to the material elastic modulus E and Poisson’sratio v. It assumes that the yield criterion f s; sy

� � ¼ 0 forDP800 satisfies the classic von Mises criterion, in which theequivalent stress is von Mises stress. The flow direction ofDp is defined by an associated flow rule with f(σ,σy). Therelation of σy and "p adopts the mechanical power fittingresults summarized in Table 1 and the plastic hardening isconsidered to be isotropic (the anisotropy has also beenproven weak and can be neglected according to the resultsprovided by Tarigopula et al [10]). The consistencycondition f

� ¼ 0 must be satisfied at any moment, so aniterative implicit Euler method is employed to acquire the Dand Cauchy stress tensor σ accurately.

The FE model for the in-plane shear specimen has beencreated respectively by plane stress elements and solidelements. The 3D model of the specimen is meshed byeight-node solid elements C3D8R with selective reducedintegration. For precision, the mesh in the shear zone isrefined with the element size of 0.25 mm×0.25 mm×0.4 mm, where 0.4 mm is the dimension of thickness in theelement; and for computational efficiency, the other areasmeshed with a relatively coarse size of 2 mm×2 mm, as canbe seen in Fig. 12. In order to be consistent with theexperimental conditions, the two clevis pins are modeled asrigid bodies at the holes of the specimen. Two referencepoints are respectively assigned to the pins, and theirtranslation degrees are coupled with those of thecorresponding pin outer surface. Contact conditions betweenthe pin outer surface and the corresponding hole innersurface of the specimen have been defined with a frictioncoefficient 0.1. In addition, the boundary conditions areapplied on the reference points, where the left point RP2is fixed and a displacement along the x1-axis is definedon the right point RP1.

The plane stress FE model for the specimen isgenerated by using the four-node elements CPS4R withselective reduced integration. The reason that the planestress element is applicable lies in the no occurrence ofnecking phenomenon in the tests. Similar to the solid FEmodel, the plane stress FE model refines the meshdistribution in the shear zone with an element size of0.25 mm×0.25 mm. The other settings for the plane stressmodel resemble those for the solid FE model.

The updated Lagrangian method is made use of tocalculate the elastic-plastic response of the in-pane shearspecimen in Abaqus/Standard. The output stress is Cauchystress, which represents the true stress state. The outputstrain is logarithmic strain and the shear strain is engineeringstrain, which equals to two times of the shear tensor straindefined by equation (4).

Comparison and Discussion

Comparisons between the experimental and numericalresults in terms of the relationships of the applied load andthe elongation and the evolutions of the shear strain within theshear zone are carried out. Detailed investigations on the shearzone including the shear stress distribution, the variations ofthe three main in-plane stress components, and the variationsof the stress triaxiality and the Lode parameter, are alsoperformed.

The curves of the applied load versus the elongationaccording to both the plane stress model and solid modelanalysis for the gauge length 95.3 mm are compared withexperimental results, as depicted in Fig. 13. The shownthree f-d curves coincide very well.

Fig. 14 shows the comparisons of the DIC testedlogarithmic shear strain distribution in the shear zone withnumerical results. From the results, we can see the shearstrain concentrates on the shear zone and the shear straindistribution band goes through a finite rotation. Thiscoincides with the observed behavior in Fig. 10. The goodagreement exhibited in Fig. 13 and 14 implies that the FEanalysis conducted in the present study is correct and wecan employ the effective numerical results to make adetailed study on the deformations of the shear zone.

The shear stress S12 distributions in the shear zone arevital in order to judge whether or not the in-plane shearspecimen has been designed successfully. The Fig. 15describes the evolutions of the shear stress S12 distributionin the butterfly shear zone with elongation, where Fig. 15(c)reflects the state at the moment close to fracture. From thecontours it can be observed that the shear stress S12concentrates on the center area of the shear zone anddistributes uniformly in its covering region. The noteworthyshear effect is exactly what is pursued.

0 3 4 5210

2

4

6

8

H31 fd

plane stress fd

solid fd

Elongation d /mm

Load

f/K

Nfracture

Fig. 13 Comparison of the curves of the applied load versus theelongation between experimental and numerical results

Exp Mech

(a) d=0.25mm

(b) d=2.00mm

(c) d=3.86mm, close to fracture

DIC result FE result

DIC result FE result

DIC result FE result

x2

x1

x2

x1

x2

x1

Fig. 14 Comparisons of theDIC tested logarithmic shearstrain distribution (plotted on theinitial geometry) in the shearzone with FE numerical results

Elastic and plastic decomposition of the rate of deformation D ¼ De þ Dp

Relation of Jaumann rate with elastic part of the rate of deformation srJ ¼ Ce : De

Plastic yield criterion f s; sy

� � ¼ s sð Þ � sy "p� � ¼ 0

Associated flow rule Dp ¼ l�@f@s ; "p ¼ l

Consistency condition f�¼ 0

Loading and unloading conditions f s; sy

� � � 0; l � 0; f l ¼ 0

Table 2 The materialconstitutive equations used inFE analysis

Exp Mech

The experimental observations in Fig. 10 and the shearstrain comparisons in Fig. 14 together show that the shearzone undergoes a finite rotation in the process of beingloaded. In the section of ‘Experimental Results,’ it ismentioned that the rotation may induce the variations of the

normal stress components in shear zone. Accordingly, thevariations of the three main in-plane stress components atthe center point of the shear zone are depicted in Fig. 16.The shear stress component climbs rapidly in the earlystage of the deformation, while it increases mildly in thelater stage until fracture. Both the normal stress S11 and S22behave with a fluctuation relative to the zero-axis. S11declines first and then climbs, while the stress S22 behavesconversely. However, both are retained at a relatively lowstate in comparison with the shear stress. The resultingeffect is in favor of the shear stress dominating the shearzone and insures that the realized simple shear stress statein the shear zone always approximates pure shear state. It isalso observed that only the shear stress exists when theelongation equals 2.0 mm and the corresponding stress stateis pure shear. This agrees with the deformed configurationof that moment shown in Fig. 15(b).

Comparisons of the experimental nominal shear stresswith numerical shear stress S12 at the center point of theshear zone are presented in Fig. 17. The nominal shearstress is the applied force divided by the nominal cross-sectional area of the shear zone; and in present work thecross-sectional area is 5 mm×2 mm, where 5 mm refers to

(a) d=0.25mm

(b) d=2.00mm

(c) d=3.86mm, close to fracture

x1

x2

x1

x2

x1

x2

Fig. 15 Evolutions of the shear stress S12 distribution in the shearzone

0 1 2 4 530.0

0.2

0.4

0.6

0.8

plane stress FE S12 solid FE S12 H31 nominal shear stress

Elongation d /mm

She

ar s

tres

s /G

Pa

fracture

Fig. 17 Comparisons of the experimental nominal shear stress withnumerical shear stress S12 at the center point in the shear zone

0 1 2 3 4 5 6

-0.2

0.0

0.2

0.4

0.6

0.8

S12

S11 S22

Elongation d /mm

Str

ess

/GP

a fracture

Fig. 16 Variations of the three in-plane stress components at thecenter point of the shear zone

Exp Mech

the nominal shear length. It is seen that the three curvesgenerally coincide well, though a small difference is observed.The fact that the actual shear length is always varying with therotation of the shear zone may be appropriate for interpretingthe difference. This can be further explained by thephenomena displayed in Fig. 15(c) that the nominal shearstress equals the numerical results when the elongation isclose to 2 mm. As mentioned above, the corresponding stateof that moment is pure shear, and the actual shear lengthequals the invariable nominal shear length.

Figure 18 shows the equivalent plastic strain distributionof the shear zone when the specimen is at the point offacture. As can be seen, the deformation is highlyconcentrated within the shear zone. As we know, theequivalent plastic strain belongs to internal variable and isclosely linked with the level of the material damageaccumulation. Therefore, within this zone the fracture ismost possibly initiated and the center point is adopted toinvestigate the damage accumulation.

The crack formation of ductile materials has been foundthat it is closely related to the endured stress state, whichcan be described by the stress triaxiality and Lodeparameter. The stress triaxiality and Lode parameter canbe formulated as

h ¼ sm s= ð5Þ

x ¼ 27

2

J3s3 ð6Þ

in which J3 is the third invariant of the stress deviators. Interms of principal components S1, S2, S3, the stress deviatoris defined by J3=S1S2S3. So the evolutions of the stresstriaxiality and Lode parameter with the equivalent plasticstain in the crack initiation are obtained and described inFig. 19. The stress triaxiality increases monotonically from

a negative to a positive value, and reaches a maximum of0.03 at the point of fracture. The obtained lower stresstriaxiality and the Lode parameter imply that the shear stateapproaches the pure shear quite closely.

The average stress triaxiality and Lode parameter [12,14, 19] are defined by

hav ¼Z "f

0hd"

� �"f

ð7Þ

xav ¼Z "f

0xd"

� �"f

ð8Þ

where η is the stress triaxiality, ξ is the Lode parameter, ‘av’means ‘average,’ " is the equivalent plastic strain and "fdenotes the " at incipient fracture. According to the data inFig. 19, the calculated ηav is equal to 0.006 and ξav is 0.03.The obtained low values of ηav and ξav indicate that thepresented in-plane shear specimen design is suitable forstudying the fracture damage under the shear stress state.

Concluding Remarks

The stress and strain relationship for DP800 is determinedby using uniaxial tensile tests and plastic hardening, whichis described by a power function. Based on the work of Bao[9] and Tarigopula [10], a different in-plane shear specimensuitable for ductile fracture studies has been developed. Theshear tests are carried out and a non-contact optical field-measuring technique, DIC, is adopted to measure theelongation of the gauge length and the shear straindistribution of the shear zone. It is shown that the shearzone of the specimen undergoes a large rotation during thetests which may induce the variations of the stress state.Furthermore, the deformation concentrates on the shearzone and no thinning phenomenon has been observed,which implies there exists no plastic instability in this kindof shear test.

Fig. 18 Equivalent plastic strain distribution of the shear zone on thepoint to fracture

0.0 0.2 0.4 0.6 0.8 1.0

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

stress triaxilityLode parameter

Equivalent plastic stain

Str

ess

tria

xilit

y an

d Lo

de p

aram

eter

fracture

Fig. 19 Evolutions of the stress triaxiality and the Lode parameterwith the equivalent plastic stain at the potential crack initiation

Exp Mech

Based on the implicit FE solver Abaqus/Standard,numerical simulations of the quasi-static shear test areconducted by using plane stress and solid elements. Amaterial model with von Mises yield criterion and isotropichardening are utilized in the calculations. It is noted that thenumerical results in terms of the relationship of the appliedload versus the elongation of the gauge length are in goodagreement with experimental results analyzed by DIC. Inaddition, based on numerical calculations, the shear zone ofthe specimen is analyzed in detail including the shear stressdistributions, the variations of the three in-plane stresscomponents, comparisons of the nominal shear stress withthe actual shear stress, the equivalent strain distributions andthe evolutions of the stress triaxiality and Lode parameter. It isfound that the shear deformation concentrates on the designedshear zone and the shear stress distributes widely anduniformly. The stress state within the shear zone is dominatedby shear stress, and the other two normal stress componentsare repressed at a relatively low level and fluctuate. Thenominal shear stress and the actual shear stress are generallyconsistent. The equivalent strain is highly located in the shearzone, where incipient fracture is most likely to occur. Theevolutions of the stress triaxiality and the Lode parameter atthe center of shear zone are provided, whose maximum andaverage values are both maintained at a low level. The aboveobtained results demonstrate the in-plane shear specimendesign presented in this work is suitable for fracture studies ofhigh strength materials under the shear stress state.

Acknowledgements The present work has been supported finan-cially by National Natural Science Foundation of China (Grant No.50775120). Thanks are also given to SSAB Swedish Steel Works forsupplying DP800 sheet metal.

References

1. Jeanneau M, Pichant P (2000) The trends of steel products in theEuropean automotive industry. Text of a conference presented atthe 55th Congress of ABM, Brazil

2. Heimbuch R (2006) An overview of the auto/steel partnership andresearch needs. report in advanced high strength steel workshop.Arlington, Virginia, USA

3. G’Sell C, Boni S, Shrivastava S (1983) Application of the planesimple shear test for determination of the plastic behavior of solidpolymers at large strains. J Mater Sci 18:903–918

4. Bouvier S, Haddadi H, Levee P, Teodosiu C (2006) Simple sheartests: experimental techniques and characterization of the plasticanisotropy of rolled sheets at large strains. J Mater ProcessTechnol 172:96–103

5. Iosipescu N (1967) New accurate procedure for single sheartesting of metals. J Mater 2(3):537–566

6. Walrath D E, Adams D F (1981). The Iosipescu Shear Test asApplied to Composite Materials. Exp Mech, 105–110.

7. Odegard G, Kumosa M (1999) Elasto-plastic analysis of theIosipescu shear test. J Compos Mater 33(21):1983–2001

8. Barnes JA, Kumosa M, Hull D (1987) Theoretical and experi-mental evaluation of the Iosipescu shear test. Compos Sci Technol28:251–268

9. Bao Y, Wierzbicki T (2004) On fracture locus in the equivalentstrain and stress triaxiality space. Int J Mech Sci 46:81–98

10. Tarigopula V, Hopperstad OS, Langseth M et al (2008) A study oflarge plastic deformations in dual phase steel using digital imagecorrelation and FE analysis. Exp Mech 48:181–196

11. Rice JR, Tracey DM (1969) On the ductile enlargement of voidsin triaxial stress fields. J Mech Phys Solids 17:201–217

12. Bao Y (2005) Dependence of ductile crack formation in tensiletests on stress triaxiality, stress and strain ratios. Eng Fract Mech72:505–522

13. Seppala ET, Belak J, Rudd RE (2004) Effect of stress triaxialityon void growth in dynamic fracture of metals: a moleculardynamics study. Phys Rev B 69:134101

14. Wierzbicki T, Xue L(2005). On the effect of the third invariant ofthe stress deviator on ductile fracture. Technical report, Impactand Crashworthiness Laboratory, Massachusetts Institute ofTechnology, Cambridge, MA.

15. Bruck HA, McNeill SR, Sutton MA, Peters WH (1988) digitalimage correlation using Newton-Raphson method of partialdifferential correction [C]. SEM spring conference on experimentalmechanics held in Portland on June 5–10

16. Zhang D, Eggleton CD, Arola DD (2002) Evaluating themechanical behavior of arterial tissue using digital imagecorrelation. Exp Mech 42(4):409–416

17. Belytschko T, Liu WK, Moran B (2000) Finite elements fornonlinear continua & structures. John Wiley & Sons

18. ABAQUS Theory Manual, Version 6.6 (2006). Hibbitt, Karlssonand Sorensen Inc., Pawtucket, USA

19. Wierzbicki T, Bao Y, Lee Y-W, Bai Y (2005) Calibration andevaluation of seven fracture models. Int J Mech Sci 47:719–743

Exp Mech