An Experimental and Numerical Study of Deformation Behavior of Steels in Biaxial Tensile Tests

Dilip Banerjee, Mark Iadicola, Adam Creuziger, and Timothy Foecke

Material Measurement Laboratory

National Institute of Standards and Technology (NIST)

Gaithersburg, MD 20899

Keywords: Cruciform, biaxial tensile tests, finite element analysis

Abstract

Lightweighting materials are increasingly being used by automotive companies as sheet metal 

components to meet fuel economy targets by the year 2025. Consequently, accurate material model data 

need to be developed by applying biaxial strain paths with cross-shaped specimens, since traditional 

uniaxial stress-strain data cannot capture the deformation behavior in complex forming operations.

This paper discusses the development of finite element models and verification of these models against 

experimental measurements. Such verification is the first step in developing procedures for making an 

optimum cross-shaped specimen design. Computed results of deformation, strain profile, and von Mises 

plastic strain in two different specimens agree with experimentally measured values along critical paths in 

the specimens. In addition, simulated results predict correctly the eventual failure location in the samples. 

Detailed analyses also suggest that specimen thickness has an influence on the eventual mode of failure. 

Further studies are being conducted to confirm this conclusion.

1. Introduction

Automotive companies are actively interested in the increased use of advanced lightweighting materials 

such as advanced high strength steels (AHSS), aluminum alloys, and fiber reinforced plastics as sheet 

metal components to meet planned fuel economy targets by the year 2025. However, accurate material 

models are needed for wider adoption of these materials in the automotive industry. These material 

models need to be validated by using complex strain paths and plastic strains as typically encountered 

during the metal forming operations. Traditional uniaxial stress-strain data are not sufficient to capture the 

deformation behavior during these complex forming operations. Such data are often developed using 

bilinear strain paths using cross-shaped biaxial specimens (i.e., cruciform) [1, 2, 3, 4].

An important issue for successful cruciform testing is the specimen design itself. Over the years, 

researchers have used a wide variety of designs [5, 6, 7, 8, 9, 10, 14]. Generally the design of the 

specimens optimizes the following goals: (a) Both strain uniformity and eventual failure in the biaxially 

loaded gauge area, (b) Reduction of stress concentration outside of the gauge area, (c) Minimization of 

shear strains in the gauge area, (d) Similar behavior during repeat tests.

A goal of the present research is to develop cruciform designs that achieve these objectives for a wide 

variety of materials. Traditional finite element analysis (FEA) in conjunction with an optimization 

software can be used to conduct a systematic study to develop an optimum design of cruciform specimens 

that fulfils the above objectives. However, the first step toward achieving this goal is to verify the FEA 

models against detailed and careful experimental measurements. Additionally, if an appropriate 

constitutive law of the material is known, the FEA analysis can be used to determine stresses in the gauge 

section of the specimen, which are often difficult or impossible to measure directly by experiments.
Recently, the NIST Center for Automotive Lightweighting (NCAL) [11] has developed a state of the art 

biaxial testing facility that can (a) impart large biaxial strains on a cruciform sample using controlled 

nonlinear and non-monotonic strain paths without unloading the sample, (b) permit strain control and 

strain measurement using a combination of non-contacting 2D real-time and 3D post-processed Digital 

Image Correlation (DIC) systems, and (c) conduct X-ray diffraction for obtaining surface stress state in 

the gauge area of interest. In this work, only small portions of these capabilities are used.

The objective of the present paper is three-fold. The first goal is to develop finite element models of 

cruciform specimens and verify these models using careful experimental measurement of strain and 

displacement fields. Toward this end, an initial design of the cruciform specimen was chosen based on a 

literature review. The second goal is to provide insight into the plastic deformation behavior of the 

specimen design in the gauge area using both experimental and simulated results. The third goal is to 

develop capabilities for systematically comparing DIC measurement results with FEA computed data.

2. Cruciform Specimen Design and Biaxial/Uniaxial Tensile Tests

2.1 Cruciform Specimen Design

The inherent advantages of using the cruciform geometry for in-plane biaxial tests are that (a) the region 

of interest is far away from locations where boundary conditions are applied (contact or friction problems 

are not typically encountered) and (b) varying strain ratios can be applied along the arms of the specimen 

(e.g. equal biaxial strain evolving into plane strain). As mentioned earlier, the designs of the cruciform 

specimens vary widely. In most of the specimen designs, failure occurred outside the central gauge 

section of interest and at strains well below traditional forming limit strains. This is due to a combination 

of various factors, most notably stress and deformation concentrations at corners or near slits in the arms 

[12]. In some of these designs, the central gauge section is thinned down to achieve higher strain and 

possibly failure in the middle of the gauge section. This is a convenient starting point for specimen design 

although one would prefer using as-received sheets in the specimen design without altering thicknesses. 

In this study, the specimen design follows that of Abu-Farha et al. [10], with the initial specimen being 

enlarged in all three directions to a thickness of 0.953 mm. Abu-Farha et al. [10] reported failure in the 

central region of the gauge section for heated specimens employing this geometry, but did not report 

results from any room temperature tests. This present specimen design is shown in Fig. 1. Cruciform 

samples were machined by water-jet with the X-axis aligned with the rolling direction of the sheet. One 

specimen was made of 9.525 mm (0.375 in) thick hot-rolled low carbon steel (hereinafter called “thick 

specimen”). The other specimen was made of 3.175 mm (0.125 in) thick cold-rolled AISI 1008 steel 

(hereinafter called “thin specimen”). This steel has much lower yield and ultimate tensile strength than 

the hot-rolled low carbon steel used for the thick specimen. Fig. 1 shows designed dimensions of the thick 

specimen. The thin specimen used a similar geometry except that it had a designed thickness of 0.749 mm 

(0.0295 in) in the center pocket (1.09 mm (0.043 in) of material was removed from top and bottom 

surfaces using electric discharge machining i.e., EDM) and that the fillet radius at the peripheral region of 

the central pocket was 0.397 mm (0.016 in). 

2.2 Biaxial Tensile Tests

Biaxial loading was applied using four hydraulic actuators, which are controlled in orthogonal pairs [12]. 

Each of these actuators has 500 kN load capacity and has a ±50 mm displacement range from a reference 

distance of 640 mm between grip faces on the X or Y-axis. Details of the loading and in situ data 

acquisition using DIC system are described in [12]. In the present study, the biaxial tests were conducted 

using displacement control. An attempt was made to apply the same 0.004 mm/s displacement rate on 

both axes simultaneously. After fracture was observed, the displacement on both of the axes was stopped 
before full separation of the test piece. Fig. 2 shows force-displacement curves in the X-direction for these 

two tests (note the force scale is substantially larger for the thick specimen). Similar curves were 

recorded for the Y-direction (not shown). 

The 3D displacement of top surface about the center of the cruciform specimen was measured with a 

stereo DIC system over a total field of view of approximately 70 mm x 65 mm. The system measures the 

surface U, V, and W displacements in the X, Y, and Z directions, respectively. Note that DIC requires a 

portion of the surface on which to correlate, thus the measurement points begin about 0.3 mm from the 

edges of the surfaces. Surface strains are calculated from the measured displacement fields. Note the 

following uncertainties in measurements: force +/- 0.025 kN; U, V, W displacements: (1.5, 1.5, and 2.5)

m respectively (thin specimen); U, V, W displacements: (3.0, 3.0, and 4.0) m respectively (thick 

specimen). 

Following the cruciform biaxial tensile tests, history-dependent U and V values at locations 

approximately 25 mm away from the central point were extracted for subsequent use in FEA simulation 

(discussed later). The displacement values were interpolated from the DIC field data at the FEA nodal 

coordinate values nearest to X=25 mm across the X-arm and nearest to Y=25 mm across the Y-arm (see 

displacement boundary locations in Fig. 3). Since symmetry was used in the model (discussed later), the 

same symmetry was applied to the sampling of the DIC data and only the average U values were used for 

the X-arm and similarly average V values for the Y-arm in the FEA model. 

As stated earlier, cross-shaped specimens tend to concentrate strain in the corners between the arms, 

which leads to failure starting in the corners or premature failure in the gauge area. Any FEA model that 

will be used for specimen optimization must be able to capture the mechanical behavior in these corners. 

In order to compare the experimental and numerical results in these critical areas, displacement and strain 

data were extracted along a diagonal line (divided in 20 equal points) that bisects the region between the 

Fig. 1 Cruciform geometry used for the thick 

specimen (dimensions in mm).

Fig. 2. Force-displacement curves for X-axis from 

cruciform specimens. Curves for Y-axis are similar 

and are not shown. Also shown points marked A 

through E, when DIC data along a diagonal path are 

extracted (see text).
central pocket and the reentrant radius at the meeting point of the X and Y arms (see "Path" in Fig. 3). 

This results in four sets of measured data, one from each corner area, for each sample. These data will be 

compared to the model predicted data along the same path. Note that these data were extracted at five 

specific times as shown in Fig. 2. The strain fields for the point of maximum force will also be compared. 

Uncertainties in strain measurement were: +/- 0.000450 m/m (in xx and yy) to +/- 0.000650 m/m (von 

Mises and zz) for thin specimen and +/-0.000400 m/m (in xx and yy) to +/- 0.000600 m/m (von Mises 

and zz) for thick specimen. Stress uncertainty varied from +/-0.45 MPa to +/-0.8 MPa. 

2.3 Uniaxial Tensile Tests

Uniaxial tensile tests were performed for both steels in the rolling direction of the sheet. These tests were 

performed using the X-axis of the cruciform machine and the DIC system. The gauge section as designed 

has a nominally parallel length of 101.6 mm and width of 19.05 mm with radii of 50.8 mm to the 28.58 

mm wide end-tabs. The same planer geometry was used for both thicknesses of material. The asmachined (by water-jet) widths and thicknesses were measured and used to calculate the initial crosssectional area. Since the biaxial test and model will achieve strains beyond the uniform strain of a 

standard tensile test, it is necessary to either extrapolate the standard test data, or approximate the truestress/true-strain relationship, to strains beyond uniform deformation. An approximation method was 

chosen that uses the DIC data sampled across the width of the specimen at the axial location of the 

eventual localized neck and failure. True axial strain was calculated as the average axial strain across the 

width for each point in time. The true stress was calculated as the applied end force divided by the 

current cross-sectional area at the same location and time. No correction for stress triaxiality was used. 

The resulting stress strain curves are shown in Fig. 4.The uniaxial test for the thin material used a constant 

displacement rate, which resulted in an undesirable increase in strain rate from approximately 8x10-5s

-1

before localization (maximum force) to 2x10-3s

-1

as the sample approached failure. In the thick material 

uniaxial test the displacement rate was reduced twice during the test (at approximately 0.26 and 0.49 true 

strain) to keep the average strain rate near 9x10-4s

-1

. Strain uncertainties varied from +/- 0.000400 m/m 

(in xx and yy) for the thin specimen to +/-0.000800 m/m (in xx and yy) for the thick specimen.

Displacement boundary 

conditions

Path

Fig. 3. A portion of the finite element mesh of the thin 

cruciform specimen. Also shown is the path in red color 

on which the FEA results are extracted for comparison 

with experimental measurements. Locations of axial 

displacement boundary conditions are also shown.

Fig. 4. True stress – true strain data obtained in 

uniaxial tensile tests.
3. Finite Element Modeling

A 3D finite element model of the cruciform specimens was developed using ABAQUS1software [13]. 

Constitutive material model data were obtained from experimental stress-strain measurements in uniaxial 

tension tests as shown in Fig. 3. Isotropic hardening is assumed in this study. Commonly used steel 

elastic properties were assumed for both materials, with a Young’s modulus equal to 210 GPa and 

Poisson ratio of 0.3. A 1/8th symmetry FEA model was developed by taking advantage of the mirror 

symmetry in the specimen shape and loading. The solid model was developed in several parts. Each part 

was meshed individually using mapped meshing. 3D linear hexahedral elements (both regular and 

reduced integration elements e.g. C3D8 and C3D8R) were used for meshing. Note that C3D8R has one 

Gauss (integration) point per element located at the centroid of the element. The C3D8 element has six 

Gauss points. Therefore, the results obtained with C3D8 elements will be more numerically accurate than 

those obtained with the C3D8R element. Mesh seeding of the parts was done such that the number and 

thicknesses of elements across any two parts is the same at the interface. This approach offers flexibility 

in controlling meshing in this geometry as the cruciform model contains very fine and thick geometric 

features. It is difficult to obtain an acceptable mesh using global seeding. Subsequently, the meshes of 

individual parts were attached and the common nodes in the interface regions were merged. Fig. 3 shows 

a typical FEA mesh of the thin cruciform specimen. Note that the finite element model includes half 

thickness of the actual cruciform specimens. Two different FEA models were developed for the thin and 

thick cruciform specimens.

For the 1/8th symmetry model, appropriate symmetry boundary conditions were applied to each of the 

three symmetry planes. On each arm of the model, the nodes that are about 25 mm away in axial 

directions were assigned history-dependent displacement boundary conditions obtained from DIC 

measurements (see section 2 and Fig. 3). Note that all nodes along the depth were assigned the same 

boundary condition as those on the surface. On the X-arm, the nodes that have different Y coordinate 

values, but the same coordinate X value, are assigned different U displacement boundary conditions as 

obtained from DIC measurements. Such an approach is also followed for V displacement boundary 

condition on the Y-arm of the model for nodes that are located 25 mm away from the central node (but 

with varying X coordinate values). The ABAQUS implicit (or standard) solution algorithm was used [13]. 

Following completion of simulation, a path was defined along the diagonal using the same points used in 

the DIC analysis (see section 2.2 and Fig. 3). Computed strain and displacement data were mapped to 20 

equidistant points along this diagonal for comparison with experimental measurements (discussed below 

in section 4). This mapping was done for each load point (squares in Fig. 2) corresponding to those when 

data were extracted from DIC measurements. 

4. Results and Discussion

In this section FEA results will be compared with data obtained from experimental measurements. These 

data will be compared along the path as shown in Fig. 3. Such data are available for 5 points as shown in 

Fig. 2. In this paper, only results at the maximum load point will be discussed in detail. Consequently, 

experimental and simulated results extracted at point C in Fig. 2 for the thin specimen and point D in Fig. 

2 for the thick specimen are discussed in this section. Also, all results are shown for the top surface of the 

model to be consistent with the DIC measurements. Overall, FEA results of displacement and strain fields 

 

1

Certain commercial software or materials are identified to describe a procedure or concept adequately. Such 

identification is not intended to imply recommendation, endorsement, or implication by NIST that the software or 

materials are necessarily the best available for the purpose.
show broad agreement with experimental data for all five points except for the first point for the thick 

sample. This is possibly due to elastic-plastic transient effects, and that the FEA model does not explicitly 

model the yield point behavior seen in Fig. 2 for thick specimen.

4.1 Displacement fields 

Figure 5(a) and (b) show both experimental and computed U and V displacement fields in the thin 

specimen. Recall that there are 4 equivalent paths for which experimental data have been measured (e.g. 1 

path for each quadrant, see Fig. 3). However, because of the symmetry used in the model, only one plot is 

shown for each ABAQUS model. Two sets of ABAQUS results are shown, one for reduced integration 

and one for regular elements. Figure 5 shows that the model predicts the displacement field accurately. 

There is variation in the experimental plots of V displacement profiles at the beginning of the path (e.g. 

near the X-arm), while the ABAQUS plots fall within the experimental curves. This could be the result of 

slight misalignment of the specimen to the X loading axis.

For the thick specimen, the results are shown in Fig. 6. Considerable variation is noticed among the 

experimental data. Overall, the computed displacement profiles are steeper than those observed 

experimentally. Both the computed U displacement and V displacement profile values are higher than the 

corresponding experimental values near the profile maxima. The reason behind this discrepancy is not

Fig. 5. Displacement along path for the thin specimen 

corresponding to the point C in Fig. 2. U-expt-1 to U-expt-4 show 

experimental U values along four equivalent paths; ABAQUS-UC3D8R and ABAQUS-U-C3D8 are ABAQUS plots using C3D8R and 

C3D8 elements respectively. Similar naming convention is used for 

V displacement plots in Fig. 5(b).

Fig. 6. Displacement along path for the thick specimen corresponding 

to the point D in Fig. 2. U-expt-1 to U-expt-4 show experimental U 

values along four equivalent paths; ABAQUS-U-C3D8R and ABAQUS-UC3D8 are ABAQUS plots using C3D8R and C3D8 elements respectively. 

Similar naming convention is used for V displacement plots in Fig. 6(b).
clear. However, a full FEA model could have captured the material behavior more accurately because 

each arm would have proper boundary conditions corresponding to the actual experimental 

measurements. In the present case, average displacement data obtained in experimental measurements 

were used in the 1/8th symmetry model, thus averaging out any misalignment between the specimen and 

the loading axes. The thick specimen model shows slightly higher displacements than the thin specimen 

model, which one would expect by comparing the force-displacement curves in Fig. 2.

4.2 Strain fields

Figures 7(a), (b), and (c) show plots of normal strains in X, Y, and Z directions for the thin specimen 

corresponding to the point C in Fig. 2. The measured thickness strains were calculated assuming volume 

conservation, using ezz = -(exx+eyy). It is obvious from these plots that the ABAQUS simulations with 

C3D8 elements are in better agreement with experimental results compared to C3D8R elements. 

However, the modeled peak ezz for C3D8 element exceeds those seen in experimental data. It is also 

noticed that the model predicts a slightly narrower width of the ezz profile than the experimental results.

For the thick specimen, the normal strain plots are shown in Fig. 8. Note that exx plots (Fig. 8(a)) show

Fig. 7. X, Y, Z normal strains for the thin specimen corresponding to the point C in Fig. 2. exx-expt-1 to exx-expt-4 show experimental Xnormal strains along four equivalent paths. ABAQUS-le11-C3D8R and ABAQUS-le11-C3D8 are ABAQUS X-normal strain plots using C3D8R 

and C3D8 elements respectively. Similar naming convention is used for Y and Z normal strain plots in Fig. 7(b) and 7(c).

Fig. 8. X, Y, Z normal strains for the thick specimen corresponding to the point D in Fig. 2. exx-expt-1 to exx-expt-4 show experimental Xnormal strains along four equivalent paths. ABAQUS-le11-C3D8R and ABAQUS-le11-C3D8 are ABAQUS X-normal strain plots using C3D8R 

and C3D8 elements respectively. Similar naming convention is used for Y and Z normal strain plots in Fig. 8(b) and 8(c).

c

c
that ABAQUS curves are within the experimental data except near the beginning of the path. Reverse 

trend is seen in for eyy plots in Fig. 8(b), where the computed data deviate at the end of the path. This is 

consistent with the discrepancy seen in plots of displacement field shown in Fig. 6(a) and (b). Fig. 8(c) 

show that computed ezz data deviate at both ends of the path but they are within the experimental bounds 

near the maximum point. This results in a sharper peak in the ezz curves of the model than in the 

experimental curves. 

4.3 Equivalent plastic strain

The von Mises equivalent strain plot obtained from DIC measurements is shown in Fig. 9. Note that this 

equivalent strain includes both elastic and plastic components, and extends to only about 0.3 mm of the 

surface edges. This experimental contour plot is compared with the von Mises equivalent plastic strain 

(PEEQ) available in ABAQUS. This comparison is justified since the expected contribution of elastic 

strain is very small.

Fig. 9 shows the measured equivalent field for the thin specimen, with largest value at the reentrant radius 

of about 0.38 and around 0.084 in the central region. Uniform equivalent strain is seen in the central 

pocket, which is also seen in the ABAQUS results in Fig. 10. The ABAQUS results show a value of 0.16 

in the central pocket and a value of 0.3 at the outer edge of the reentrant corner. The model shows a slight 

decrease in strain along the diagonal from the reentrant radius to the peripheral region of the central 

pocket, where a value of about of 0.24 is obtained. Although the model does not show as thick a band of 

PEEQ of 0.3 along the diagonal from the reentrant radius to the peripheral region of the central pocket (as 

seen in experimental result in Fig. 9), it is intuitive that the failure initiation will probably start at the 

strain concentration at the base of the fillet from the thicker region into the thinner region. In fact, the 

experiment showed that the failure initiated at both the top left and bottom right of the pocket area at the 

base of the fillet radius, and these cracks propagated from the periphery to the center of the pocket.

For the thick specimen, the measured von Mises equivalent strain contour plot is shown in Fig. 11. It is 

clear from this plot that strain values are higher along the diagonal, which is at 45 ̊ to the horizontal than 

the corresponding values along the diagonal that is at 135 ̊ to the horizontal. Along this diagonal (at 45 ̊

to the horizontal), equivalent strains of 0.27 or greater are seen at the peripheral region of the central 

pocket. A small band of 0.27 equivalent strain is seen near the outside edge of reentrant radius. ABAQUS 

computed results are shown in Fig. 12 (for the same load level). The ABAQUS strains range from 0.38 in 

Fig. 9. Measured von Mises equivalent strain in 

the thin specimen at the maximum force point.

Fig. 10. ABAQUS computed von Mises equivalent 

plastic strain in the thin specimen at the maximum 

force point.
the central region of the pocket to about 0.25 near the peripheral region of the center pocket. An 

equivalent strain value of 0.24 is seen at the reentrant radius in the model. It is conceivable that failure 

will occur in the central pocket. In the actual experiment, the thick sample cracked in the central pocket 

region away from the periphery.

From the analysis of these two tests, it appears that that the thickness of the sample does have an 

influence on the eventual mode of failure. However, it is not possible to make a definite conclusion at this 

time because these two cruciform specimens were not made from the same material and did not have the 

same processing history.

5. Summary and future work

Finite element analyses of biaxial tensile tests using cruciform specimens were conducted using 

ABAQUS software. The numerical model uses constitutive material model data obtained from uniaxial 

tests. The FEA model employed history-dependent displacement boundary condition data that were 

obtained from in situ DIC measurement in the cruciform test frame. Overall, the numerical model showed 

reasonable agreement with experimental measurements when both displacement and strain fields were 

compared. Biaxial straining were seen in the central region of interest, as desired. The model predicts the 

path along which the eventual failure was seen in the experiment. ABAQUS results using C3D8 elements 

showed better agreement with experimental data in comparison with those obtained with the reduced 

C3D8R elements. This is especially true for the thin specimen.

Future work will include: (a) developing a full model of the specimen (ignoring mirror symmetry) and 

repeating the simulations using appropriate boundary condition data, (b) including anisotropic material 

properties and other relevant constitutive models, and (c) using ABAQUS in conjunction with an 

optimization software to develop the most optimum cruciform specimen design, (d) studying the 

influence of EDM on material properties in the thinned down central region, (e) investigating the 

influence of the presence of any discontinuities or defects on the overall deformation behavior, and (f) 

investigating the influence of sheet thickness on the mode of eventual specimen failure.

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Fig. 11. Measured von Mises equivalent strain in 

the thick specimen at the maximum force point.

Fig. 12. ABAQUS computed von Mises equivalent plastic 

strain in the thick specimen at the maximum force point.
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