. . . . . 85 . . . . .

International Deep Drawing Research Group 

IDDRG 2009 International Conference 

1-3 June 2009, Golden, CO, USA 

METHOD FOR LOCAL MEASUREMENT OF STRESS AND STRAIN IN A 

FORMED PART UNDER LOAD 

M. A. Iadicola1, T. Foecke1, and L. Ma1

1

 NIST Center for Metal Forming 

Metallurgy Division, National Institute of Standards and Technology 

100 Bureau Drive, Stop 8553, Gaithersburg, MD 20899-855 USA 

e-mail: mark.iadicola@nist.gov, Web page: www.nist.gov/msel/metallurgy/materials_performance/ 

ABSTRACT 

The sheet metal forming industry requires prediction of complex deformation processes for 

various materials and alloys. The use of numerical modeling has been very successful in this 

regard for well understood and characterized materials being deformed within the characterized 

envelope. Unfortunately, new materials that require more complex constitutive laws (or are 

sensitive to particular deformation modes) are not as easily modeled, and in many cases the 

necessary experimental results do not exist or are quite limited (and do not cover the range of 

expected deformations, such as high strain under multi-axial loading). 

At the National Institute of Standards and Technology (NIST) Center for Metal Forming 

common test methods (e.g., Marciniak ram test) are being combined with in situ X-ray 

diffraction and full field digital image correlation to determine the local strains and associated 

stresses during plastic deformations. This new experimental approach can be used to more 

extensively probe the mechanical response of metal sheets being deformed in multi-axial tension. 

This paper is intended to introduce the method and present some illustrative examples. Three 

applications are presented: (1) yield surface determination, (2) benchmark testing, and (3) 

measurements near a physical feature (in this case asymmetrical expansion of a hole). It is 

shown that stress-strain behavior can be measured along multi-axial paths, and how local stresses 

near inhomogeneities can be measured and tracked in situ with increasing levels of deformation. 

Keywords: Forming; Stress Measurement; Yield Surface; Flow Stress; Multi-axial; X-ray 

Diffraction; Simulation; Standard Tests.
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M. A. Iadicola, T. Foecke, and L. Ma 

1. INTRODUCTION 

The sheet metal forming industry requires prediction of complex deformation processes for 

various materials and alloys. For years, experience with familiar materials (e.g., mild steel) 

permitted empirical correction to predictions to achieve acceptable results, but with the increased 

rate of alloy development and the decreased time to production (i.e., a year or two of experience 

rather than decades) more efficient methods are necessary. The use of numerical modeling has 

been very successful in predicting the behavior for a well understood and characterized material 

being deformed within the characterized envelope. Unfortunately, new materials that require 

more complex constitute laws (or are sensitive to particular deformation modes) are not as easily 

modeled, and in many cases the necessary experimental results to cover the range of expected 

deformations (that is high strain under multi-axial loading) are quite limited or do not exist. 

Various methods of overcoming this problem have been or are being developed (e.g., cruciform 

testing, benchmark testing, or inverse problem analysis), however these often have to assume a 

constitutive law and just compare with resulting strains. These methods are further complicated 

by stress concentrations (e.g., in reentrant corners), friction, and bending, which requires more 

elaborate modeling and assumptions before the comparison is performed. 

At the National Institute of Standards and Technology (NIST) Center for Metal Forming 

(CMF) common test methods (e.g., Marciniak ram test) are being combined with in situ X-ray 

diffraction (XRD) and full field digital image correlation (DIC) to determine the local strains and 

associated stresses during plastic deformations. This new, albeit complicated, experimental 

approach permits determination of multi-axial stress-strain response (to large strains) for various 

sheet materials without the need for a finite element analysis (FEA) with an assumed constitutive 

law. The goal is to improve the FEA through better constitutive law selection and calibration. 

This paper is intended to introduce the method and present some illustrative examples. After 

a brief description of the materials to be used for testing, the general experimental procedure 

associated with Marciniak testing, XRD, and strain measurement is described. Finally, three 

applications are presented: Example 1 - yield surface determination, Example 2 - benchmark 

testing, and Example 3 - measurements near a physical feature (in this case asymmetrical 

expansion of a hole). 

2. MATERIALS 

In this work, results for aluminum and steel alloys will be discussed, but the methods 

presented are generally applicable to many common engineering metals. These specific materials 

are chosen for their interest in the sheet metal forming community. Standard test methods are 

used to measure the static uniaxial mechanical response of the materials in the typical three 

directions (rolling direction, RD, transverse to the rolling direction, TD, and on the 45o

 diagonal 

to the rolling direction, DD). Different materials are used in each application: AA5754-O for 

Example 1; AA6022, AKDQ, HSLA50, and DP600 for Example 2; and a different HSLA50 for 

Example 3. Table 1 lists the basic material properties including: 0.2 % offset yield stress (YS), 

ultimate tensile strength (UTS), uniform elongation (UE), thinning ratio (r), and the parameters 

(n and K) for a simple power-law hardening model (σ = Kε

n

) relating strain (ε) to stress (σ). 

Note that AA5754-O exhibits the Portevin-Le Chatelier (PLC) effect, and both HSLA materials 

exhibit yield point behavior. Uniform elongation for both of these alloys has been redefined (to 

ignore the continuous localization of the PLC effect and the initial localization of the HSLA) to 
. . . . . 87 . . . . .

M. A. Iadicola, T. Foecke, and L. Ma 

be at the overall maximum of the smoothed engineering stress-strain curve during the later stages 

of hardening (see [1-2] for further details). 

Table 1. Material properties for all materials discussed in Section 4. Data (and available uncertainties, 

shown in brackets) are taken from [1-3] for all but 1.00 mm thick HSLA50. 

Material 

(thickness) Direction YS (MPa) 

UTS 

(MPa) 

UE 

(%) r 

K 

(MPa) n 

RD 94.1 [3.3] 226 [3] 18.9 [0.9] 0.69 [0.02] 474 [9] 0.317 [0.010] 

TD 92.1 [0.7] 218 [1] 21.5 [0.5] 0.87 [0.02] 458 [6] 0.326 [0.006] 

AA5754-O 

(1.00 mm) 

DD 90.9 [0.6] 216 [3] 22.4 [0.9] 0.73 [0.03] 447 [3] 0.323 [0.004] 

RD 136.0 256.9 22.2 1.029 479.9 0.258 

TD 127.6 238.3 24.0 0.728 442.5 0.258 

AA6022 

(1.00 mm) 

DD 131.2 247.6 24.8 0.532 455.0 0.254 

RD 158.3 315.0 26.4 1.546 579.9 0.256 

TD 166.0 312.0 24.6 1.942 569.0 0.250 

AKDQ 

(1.00 mm) 

DD 164.7 317.0 25.1 1.508 579.7 0.252 

RD 394.3 463.7 16.4 0.581 770.0 0.187 

TD 427.7 466.0 17.5 1.013 758.7 0.176 

HSLA50 

(0.79 mm) 

DD 395.3 447.0 17.0 1.166 733.0 0.182 

RD 420.0 688.7 14.0 0.821 1080.7 0.152 

TD 425.7 697.0 13.5 0.969 1090.3 0.151 

DP600 

(0.98 mm) 

DD 427.7 690.7 12.8 0.915 1078.2 0.150 

RD 364 [5] 427 [5] 19.1 [0.5] 1.03 [0.05] 597 [18] 0.105 [0.010] 

TD 392 [5] 435 [5] 17.5 [0.5] 1.36 [0.10] 594 [14] 0.093 [0.010] 

HSLA50 

(1.00 mm) 

DD 376 [7] 428 [5] 18.2 [0.5] 1.33 [0.12] 568 [19] 0.081 [0.010] 

3. EXPERIMENTAL PROCEDURE 

3.1 Marciniak testing 

In addition to the standard uniaxial deformation used in the previous section, this work uses 

an augmentation of the Marciniak flat ram test [4] to deform the specimen in multi-axial tension. 

The method uses as-received sheet metal samples that are trimmed down to specific widths to 

impose specific nearly linear strain ratios [5]. Figure 1 is a top view and through-section of the 

axisymmetric die, binder, and ram (with a center recess). Square samples of approximately 305 

mm on each side result in equi-biaxial (EB) deformation, and very thin strips deform uniaxially. 

Strain ratios between these two extremes (e.g., plane-strain, PS) are achieved by reducing the 

widths from the square sample to the thin strip, resulting in reduced straining in the reduced 

width direction. A reinforcement washer (typically made of a mild steel sheet with a centered 32 

mm hole) is used to reinforce the sheet along the bend of the ram and die radii, thus forcing 

deformation in the center of the sample above the center of the ram. Vegetable based oil 

lubricant is applied to the top surface of the sample where it contacts the die and the bottom 

surface of the washer where it contacts the ram bead. Columns and additional rings, not shown in 
. . . . . 88 . . . . . 

M. A. Iadicola, T. Foecke, and L. Ma 

Figure 1, reinforce the die ring. This leaves the top of the specimen (above the center of the ram) 

accessible for the metrology that will be described in the following sections. A lock-bead (see 

Figure 1) is used in all of the testing presented here. Procedurally, after specimen and washer 

insertion between the die and binder (Figure 1, through-section), the die is lowered into place 

and the binder is clamped to a selected force (setting the lock-bead, before the ram is positioned 

to lightly touch the sample). Ram displacement is controlled throughout the test, where the ram 

may be displaced continuously or to selected levels (interrupted testing) and held there for hours. 

Ram and clamp loads and positions are continuously monitored during the tests. 

Figure 1. Top view and through-section (with specimen and washer inserted) of axisymmetric ram, 

binder, and die (dimensions shown in mm). 

3.2 Strain measurement 

Two methods of strain measurement are employed in the examples that follow: mechanical 

extensometry and DIC. Early work (Example 1) used a specially made miniature biaxial 

mechanical extensometer (with a 12 mm by 12 mm footprint on the top face of the sample). This 

device is designed to travel with the top surface of the sample (attached by elastic bands) and 

permits continuous strain measurement (at Class C or better accuracy, ASTM E83-92, 

throughout the 40 % strain range) in two orthogonal directions averaged over the gage length 

between the extensometer knife-edges. Layout lines drawn on the surface of the sample permit 

alignment of the axes of the extensometer to the RD and TD of the sample. Unfortunately, the 

device can be easily bumped during other surface measurements being made during a test, 

therefore in Examples 2 and 3 the extensometer is replaced by a non-contacting commercial 3-D 

DIC system. 

The 3-D DIC system uses two cameras to acquire stereo-images of the top surface of the 

sample. Through 3-D photogrammetry, the displacements of a random surface pattern (painted 

black dots on a white base coat) are tracked from the reference image pair to the current image 

pair in all three dimensions. These displacements are then used to calculate full-field surface 

strains. Strains at specific spots of interest on the surface are then interpolated from these data. 

To synchronize the measured strains to the test, the DIC system incorporates an analog to digital 

data acquisition system that acquires the control signals of the machine at the time of image 

acquisition. Calibration of the system is checked frequently using a series of images taken of a 

reference target in various positions. Precision is dependent on multiple image processing 
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M. A. Iadicola, T. Foecke, and L. Ma 

parameters, but in general the strain resolution in the examples shown here is on the order of 200 

micro-strain to 400 micro-strain. In Examples 2 and 3, small areas on the surface of the sample 

do not have a painted pattern. These exposed areas are required for the XRD measurements (to 

be described next). The smoothness of the strain fields is verified with other fully painted 

samples, and permits confident approximation of the strains at the masked points by interpolating 

from the neighboring areas were data is available. 

3.3 X-ray diffraction measurements 

XRD is performed on a small spot on the surface of the sample during testing to measure the 

interatomic lattice spacing, and through calibrated X-ray elastic constants (XECs), determine the 

macroscopic stress in the sample. This method assumes that the material has been sufficiently 

plastically deformed to remove any residual stress from earlier processing (e.g., skin pass 

through thickness residual stress). The measurement technique is based on methods developed 

to measure residual stresses [6], and is reviewed briefly here. Bragg’s law relates the lattice 

spacing (d) of a crystalline material to the wavelength (λ) of the source X-rays and the reflected 

angle (2θ), therefore a change in lattice spacing will result in a change in the angle of reflection 

(for all other parameters held fixed). With multiple crystallites in the reflecting volume of 

material, multiple d-spacings are measured simultaneously over a range of reflection angles. 

These measurements form a distribution of intensity for changes in the reflection angle. This 

distribution is commonly referred to as a diffraction peak, and the top portion of this peak is fit to 

an appropriate function. By measuring the change in the diffraction peak position for a specific 

family of lattice planes of interest ({hkl}) the change in lattice plane spacing can be monitored. 

If the reference stress-free lattice spacing (do) is known (usually measured on a fine powder of 

the same material), then a lattice strain can be calculated which relates to the stress in the crystal 

through XECs. A key point to understand is that the crystal lattice experiences elastic strains 

only. Plastic strain will cause increased dislocation density and dislocation motion, which can 

broaden the peak, but will not change the peak position. 

Figure 2a defines the axes for the specimen surface system (Xi) and the X-ray beam and 

detector system ( Xi

ˆ ) where 3 Xˆ is the bisector of the incident and reflected beams (Figure 2b). 

The rotation angles between the systems are φ (in-plane rotation angle) and ψ (tilt from the 

surface normal, X3, about the 2 Xˆ axis). The standard definition of the average lattice strain in 

the 3 Xˆ direction is 

ο

φψ ο

d

d d

e −

33′ = , (1) 

where dφψ is the lattice spacing at a specific pair of rotation angles. After rotation of the lattice 

strain to the surface system Equation 1 in terms of surface strain would be 

cos cos sin(2 ) sin sin(2 )

cos sin sin(2 )sin sin sin

13 23

2

33

2 2

22

2

12

2 2

11

ε ψ ε φ ψ ε φ ψ

ε φ ψ ε φ ψ ε φ ψ

ο

φψ ο

+ + +

= + + −

d

d d

 (2) 
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M. A. Iadicola, T. Foecke, and L. Ma 

If an isotropic constitutive relationship is assumed for the lattice (as is typical in this type of 

analysis) the surface strain components can be replaced by stresses with the introduction of two 

XECs S1 and S2

 

{ } 2

1 { cos sin }sin(2 ) 21

{ cos sin(2 ) sin }sin

2

1

2 13 23 2 33 1 11 22 33

2

33

2

12 22

2

2 11

σ φ σ φ ψ σ σ σ σ

σ φ σ φ σ φ σ ψ

ο

φψ ο

+ + + + + +

= + + − −

S S S

S

d

d d

 (3) 

The constants S1 and S2 are dependent on material composition, lattice direction (hkl), and other 

microscopic effects, and can be calibrated experimental or occasionally found in the literature 

[6]. The XECs used in the examples to follow are calibrated experimentally from material in the 

same batch, and will be referred to as effective X-ray elastic constants (EXEC) for clarity. 

(a) (b)

Figure 2. XRD axis definition: (a) for surface, Xi, and detector, Xi

ˆ , systems; and (b) simplified case for 

φ = 0 showing 3 Xˆ as bisector of reflected beam angle for right-side detector. 

In addition to the EXEC and do, the XRD measurement of stress at any given stress state 

requires measurement at a proper variety of angle pairs to solve for the six unknown stresses. 

Over-sampling is preferred to permit an analysis of uncertainty. Certain assumptions can quickly 

reduce the number of unknowns, and thus reduce the required number of sampled angle pairs. 

Here are two cases using some common assumptions. For Case 1, if we assume that σ33 = 0 for a 

thin sheet being deformed by in-plane tension, then the stress-free lattice spacing (do) can be 

calculated from the measured data (reducing Equation 3 by one unknown). Additionally, for 

Case 2, if we assume that the principal stresses align with the principal strain axes for rolled 

material loaded in the RD or TD, then only measurements at φ = 0o

 and φ = 90o are needed to 

determine the in-plane stresses. Case 1 is assumed for Examples 2 and 3, below, where a range 

of φ-angles and ψ-angles are sampled to solve for the remaining five components of stress. Case 

2 is used for Example 1, below, where φ = 0o

 or φ = 90o and a range of ψ-angles are scanned to 

measure σ11 and σ22 only. 

For the aluminum samples, a Co X-ray source is used on the as-received sample surface. For 

all of the steels tested, a Cr X-ray source is used, and the surface has the zinc coating removed. 

The paint used for DIC can degrade the X-ray signal and affect the focus distance, therefore the 
. . . . . 91 . . . . .

M. A. Iadicola, T. Foecke, and L. Ma 

area for XRD was masked from painting and results in a small gap in the strain field data. As 

mentioned in the Section 3.2 the smooth strain fields allow interpolation through these areas. 

The size of the XRD measurement area on the surface is set by selection of an appropriate 

aperture for the end of the beam collimator (for Example 1: 5 mm by 1.5 mm, for Example 2: 2 

mm diameter and for Example 3: 1 mm diameter). 

4. RESULTS AND DISCUSSION 

In this section, three examples using the unique measurement methods described above are 

presented. In each case a brief background and description of the experiment involved is given, 

followed by a description of some of the particulars of the procedure, and culminate with 

highlights from the results. 

4.1 Example 1: Yield surface determination 

Traditional methods of multi-axial sheet metal testing (namely cruciform or uniaxial widesheet testing) require FEA to determine the stress, based on the measured load and a correction 

factor. The method described here (and in greater detail in [1]) is intended to determine these 

data less ambiguously by not requiring the assumption of a plastic constitutive law a priori. The 

use of the Marciniak testing can achieve large multi-axial strains (EB to above 20 % and PS to 

almost 15 %) well above that of the other methods (e.g., < 5 % for cruciform). 

The procedure follows that described in Section 3.1, with interrupted straining by stepping 

the ram at quasi-static rates (0.1 mm/s) followed by a hold under load during XRD focusing and 

scanning (taking about 8 minutes total time for each step in the ram). XRD scanning includes 26 

ψ-angles between -37.4o and +37.4o in one φ-angle direction (either RD or TD). This method of 

testing requires two tests for each strain path (i.e., EB, PS, etc.), one in RD and one in TD. In 

addition, each individual test was repeated (by different system operators, about one month 

apart) to verify the repeatability of the measurements. By tilting in only one direction during 

each test, the mechanical extensometer could be attached near the XRD head parallel to the tilt 

direction, but just beyond the diffraction spot. The strain measured by the extensometer is 

averaged during each position hold, and is seen to vary by <0.005 % strain during the XRD scan. 

EXEC are determined from uniaxial testing with a series of unloads [7]. Results of uniaxial 

XRD measurements are compared to the results from continuous uniaxial testing in Figure 3a for 

the RD. The agreement is seen to be excellent. 

Figure 3b plots the true stress vs. true strain results for EB, PS, and uniaxial testing using the 

XRD system for the RD direction. The measured uncertainties (approximately the size of the 

data symbols in Figure 3b) are based the quality of the linear fit of the sin2

ψ term in Equation 3, 

and are about the same amplitude as the variation of the stress due to the PLC effect as seen in 

the hardening curve of Figure 3a. From these data (and similar for tests performed in the TD 

direction), the stress values at equal plastic work levels (equivalent to the work at 1 %, 5 %, 10 

% and 15 % uniaxial RD strain levels) are calculated for all three strain paths in RD and TD. 

These results are plotted in Figure 3c along with two fits to an eight-parameter model (Yld2000-

2d, [9]) based on (1) the measured strain ratio r-values or (2) using optimized r-values based on 

the 15 % data points. Even using this rather advanced model the fit to the data does not capture 

the PS behavior without using optimized (artificially high) r-values. Additionally, the fits do not 

capture the changing EB behavior between the 1 % and 15 % levels. These seemingly peculiar 

large EB stress values, with stress ratios not equal to 1, are in a range of strain rarely achieved by 
. . . . . 92 . . . . . 

M. A. Iadicola, T. Foecke, and L. Ma 

other test methods (except pressurized tube tests). One such tube test method [9] shows similar 

behavior, also with a stress ratio >1. 

(a) (b) (c)

Figure 3. Example 1 results (from [1]) (a) comparison of continuous uniaxial test with XRD data, (b) 

multi-axial true stress-true strain data for RD, and (c) equal work data with Yld2000-2D yield curves 

based on measured r-values (Fit 1) and optimized r-values (Fit 2). 

4.2 Example 2: Benchmark testing 

The third benchmark of Numisheet 2005 conference “Channel Draw/Cylindrical Cup 

Benchmark” [10] includes a two stage forming process and uniquely requires the measurement 

and prediction of in situ stress, in addition to surface strains. Stage 1 is a channel draw process 

with variable draw bead depth, resulting in a cyclic bend/unbend deformation followed by plane 

straining in the sheet TD. This results in a sizable springback in the side-walls. Stage 2 (which 

will be discussed in this section) cuts curved samples out of these side-walls and subjects them to 

plane strain by the method described in Section 3.1 to specified ram heights, at which point the 

position is held and sample stress is measured above the center of the ram by XRD. Another 

interesting aspect of the benchmark is that the Stage 2 samples are oriented in two directions: 

Shape A approximates PS at a right angle to the plane straining of Stage 1, and Shape B 

continues the PS deformation in same direction as Stage 1. The original 2005 benchmark 

required results [11-12] and predictions for one bead depth for four materials (the AA6022, 

AKDQ, HSLA50, and DP600 in Table 1). The extended benchmark (completed recently) 

required these same measurements for four bead depths and as-received samples for all four 

materials [13-14]. Unfortunately, the AA6022 had aged too long and its results are no longer 

considered true to the original benchmark design. The details of the testing are beyond the scope 

of this work (and may be found in [14]), but some of the more interesting results are described 

here. The benchmark testing requires that a minimum of three tests be performed for each 

condition to check variability of the results. Therefore, in excess of 100 tests were performed, 

with only a very small subset described here. 

Nominally 254 mm by 135 mm samples are cut from the curved side-walls of the Stage 1 

channel draw samples after springback. Shape A samples are cut perpendicular to the curvature 

of the sheet, and Shape B samples are cut parallel to the curvature of the sheet [12]. Flattening 

of the these samples is part of the benchmark process, and is accomplished by closing of the die 

(Figure 1) on the sample prior to clamping (to set the lock bead). After clamping, the ram is 

lightly preloaded (0.2 kN) and then raised (at 1 mm/s) to a benchmark specified level slightly 

below failure [12] and held at that position. The XRD head is focused and a series of 14 ψangles at 4 φ-angles are scanned, in less than 15 minutes. These data are used as described in 

Section 3.3 to determine do and the five stress components of interest. EXEC are determined for 
. . . . . 93 . . . . .

M. A. Iadicola, T. Foecke, and L. Ma 

each material by performing elastic 4-point bend tests on a 135 mm by 20 mm sample that is 

instrumented with a calibrated foil strain gage. A minimum of five loading and unloading points 

are used for each calibration curve. 

The results of the testing showed higher than normal variability in strain and stress between 

samples for certain test conditions and certain alloys. For instance, the strain results for three 

repeated Shape A AKDQ samples for each of the three highest draw-bead depths show much 

higher variability (averaging > 2.5 % strain) than the other AKDQ samples (averaging < 1 % 

strain) and other steels (typically < 1 % strain). Similarly large variations are seen in the 

measured stress results. Recall that, in this case, the strain history of the samples has a 

perpendicular path change from Stage 1 to 2 testing. The result is a localization of the 

deformation, which can have a large effect on the measured stresses and strains. Figure 4 shows 

the first principal true strain map for three AKDQ samples (Shape A as-received, Shape A Stage 

1 bead depth penetration of 75 %, and Shape B also at Stage 1 bead depth of 75 %). The 

localization (neck) is clearly visible in Figure 4b (and when the sample is visually inspected after 

testing). Not surprisingly, stress measurements are location sensitive near a feature that results in 

a large gradient in stress. Another interesting case is the large variability in the stress results for 

HSLA samples for Shape A, which on average are twice that of the HSLA Shape B samples (for 

all the Stage 1 conditions). Figure 5a and 5b are perspective plots of two HSLA samples both 

from the same Stage 1 draw bead penetration depth, but cut down to Shape A and B, 

respectively. A periodic roughening is clearly visible, and can be felt on the part after testing. 

Stress measurement in this oscillating strain field will be quite position sensitive. 

(a) (b) (c) 

Figure 4. First principal true strain fields for AKDQ samples (a) Shape A as-received (b) Shape A 75 % 

bead depth and (c) Shape B 75 % bead depth. 

(a) (b) 

Figure 5. Perspective plot of DIC measured surface heights for HSLA 50 samples for 50 % Stage 1 bead 

depth: (a) Shape A and (b) Shape B. 
. . . . . 94 . . . . . 

M. A. Iadicola, T. Foecke, and L. Ma 

The testing method’s sensitivity to these features is seen as a testimony to its overall 

sensitivity and applicability. To show this point more clearly, a test with a specific physical 

feature with a known location is of interest. This idea leads into the next example. 

4.3 Example 3: Asymmetric hole expansion 

Our final example is a preliminary look at a more complex test. A physical feature is 

considered that will simultaneous produce a variety of strain paths and a stress field with 

measurable variation. The same Marciniak method as described in Section 3.1 is used here 

except a small (6.35 mm diameter) hole is drilled 12 mm off the center of the 305 mm x 305 mm 

HSLA sheet sample in the RD (see Figure 6a). The hole edge is deburred and sanded smooth 

after machining (similar to the washer hole, dashed line in the figure). In Figure 6a, the points 

(A, B, and C) located at the origin and at 5 mm offsets along the RD are the locations used for 

stress measurement. Similar to Example 2, the spots used for XRD have the zinc coating 

removed and are masked from the DIC paint pattern leaving small (2.5 mm diameter) areas 

lacking strain measurement (Figure 6b). After a preliminary test to failure, five ram heights are 

selected for position holds and XRD measurements. A similar procedure to the earlier examples 

is followed with an added step of repositioning and refocusing to each of the three measurement 

points (A, B, and C). Figure 6b is the strain field plot measured during the third position hold. A 

comparison of the strain fields between the XRD test (with holds) and the preliminary 

(continuous rate) test, shows the field measured with the missing data (at A, B, and C) is 

consistent with the field measured for the preliminary test. 

(a) (b) (c)

Figure 6. Example 3: (a) drawing of the geometry (dimensions in mm), (b) measured εy strain map for 

hold position 3, and (c) FEA calculated strain map at a similar strain level (plotted in same gray scale). 

An ABAQUS/Standard FEA of the test is performed using the dimensions shown in Figure 

6a. Only half of the part (TD positive) is modeled with symmetry conditions along the RD axis. 

Four node plane-stress elements (1105 elements with 1173 nodes) with bilinear interpolation are 

used. The material is modeled with a combined elasto-plastic formulation. The elastic behavior 

is treated as isotropic with a 209 GPa elastic modulus and 0.28 Poison’s ratio. The plastic 

hardening curve is taken directly from the uniaxial RD data with minor adjustment to the data at 

the yield point and subsequent plateau (this region is made positive definite, with a slope of 507 

MPa, to prevent numerical complications). The plastic anisotropy is modeled using the Hill’48 

formulation (based on the r-values given in Table 1). The loading is applied as a uniform radial 

displacement, as shown in Figure 6a, in steps chosen to approximate the experimental results for 

εY along the X-axis near A, B, and C. 
. . . . . 95 . . . . .

M. A. Iadicola, T. Foecke, and L. Ma 

Figure 6c shows the FEA predicted strain field for a displacement level similar to the test 

point shown in Figure 6b. Qualitatively, the results are consistent, and the location of the initial 

plastic behavior around the hole is consistent between the model and test. Unfortunately, the 

model prediction for step 5 shows localization of strain near the hole that is not present in the 

measurements, but the location along the hole is consistent with the area of eventual failure seen 

in the experiment. Therefore, the model can capture much of the test results, however a 

quantitative comparison of the FEA and measured results is not as favorable. Figure 7a and 7b 

are plots of the true strain (εY and εX, respectively) variation along the X-axis (Figure 6a) for 

steps 1 through 4. Uncertainties of the DIC data are smaller than the symbols used. The three 

gaps (outside of the hole region) in the measured strain data are at the points A, B, and C. The 

FEA and experimental results match well in the region of points A, B, and C (Figure 7a) since 

this is the data used to define the FEA steps, but the far field strains do not necessarily match 

(particularly X < -10 mm). The εX data along this same axis (Figure 7b) also captures the correct 

trends, but the values of true strain differ from the FEA and experiment (again greatest in the far 

field). Although a comparison of the stresses show they are off by an even greater amount, the 

FEA and measured stress values are all tensile and properly capture the spatial trends between 

points A, B and C. This quantitative disparity in stress might partially be the result of a recent 

experimental issue with the testing machine, which shows a loss of pressure at the hold points 

(unloading the sample by 3 %). These technical issues are currently being addressed, but 

assuming the small unloading results in a small (uniform) elastic unloading, the measured stress 

ratio may still be useful. A comparison of the measured (points) and FEA predicted (lines) stress 

ratio (σY/σX) is shown in Figure 7c. Although the general decrease of the stress ratio away from 

the hole is captured, the expected step-to-step variation is not correct. Steps 1 and 2 in the range 

of yield point behavior seem to follow a single trend with measured stress ratios generally 

greater than the FEA calculated ratios, and steps 3 and 4 follow the reverse trend with measured 

ratios generally below the FEA calculated values. The results at point A do not necessarily 

follow these trends, but also have the largest associated uncertainties, being in a region of severe 

stress gradients. Not surprisingly, application of the method in regions with high gradients might 

require some additional analysis. If the measured stresses are low due to the unload, increasing 

both stress values will bring the ratios closer to unity and closer to the FEA results for steps 1 

and 2, but further from the FEA results for steps 3 and 4. Therefore more investigation after 

correction of the technical issues is required to determine if there is a disproportionate unloading 

occurring in the test results presented here, or if there is an error in the FEA predicted stresses 

(and thus a problem with the assumed material model). 

(a) (b) (c)

Figure 7. Stress and strain variation along the X-axis for Example 3: (a) εY, (b) εX, and (c) stress ratio 

σY/σX. FEA results are shown as lines and measured data as points. 
. . . . . 96 . . . . . 

M. A. Iadicola, T. Foecke, and L. Ma 

5. CONCLUSIONS 

In this paper we have detailed newly developed experimental capabilities that can be used to 

more extensively probe the mechanical response of metal sheets being deformed in multi-axial 

tension. Not only can stress-strain behavior be measured along multi-axial paths, but also local 

stresses near inhomogeneities can be measured and tracked in situ with increasing levels of 

deformation. It is believed that further development of these techniques will lead to both new 

standard tests for sheet behavior and a better fundamental understanding of multi-axial forming 

behavior. 

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