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A custom biaxial testing fixture
was used to evaluate new cruciform
geometries. Specimens consisting of
AA5083, Mg AZ31B, and TWIP steel
were quasi-statically deformed to failure
at 300ºC. We elucidate geometric
differences between specimens that
accumulate plastic deformation within
their gauge areas and those that
prematurely fracture. Strain fields are
computed with digital image correlation
for selected geometries.
INTRODUCTION
Tensile testing has been a predominant
means of measuring material
strength since the Renaissance. In
fact, it was none other than Leonardo
da Vinci, who when faced with the
problem of selecting metal wire strong
enough to hang his now priceless
paintings, conceived of a simple but
elegant experiment. An iron wire, suspended
from a support, was stretched
by adding sand to a basket attached to
the free end of the wire. Once the wire
broke, its strength could be inferred
from the amount of sand in the basket.
da Vinci’s wisdom put experimental
mechanics on a path to becoming a
formal discipline with tensile testing
at its foundation.1 Since da Vinci’s
time, the strength of just about every
known engineering material has been
measured in tension. Testing technology
continues to advance with the development
of ingenious devices that
now provide tensile properties down
to molecular length scales. Miniature test systems and their application to
biological materials and nanocrystalline
metals were described in a recent
JOM article.2
There are many examples where
metals are subject to multiaxial straining.
Of these, sheet forming operations,
which provide a myriad of components
for the appliance, food, automotive,
and aerospace industries represent
a multibillion dollar industry in
the United States.3 Finite element (FE)
simulation is now an accepted means
for aiding forming processes and component
designs, but accurate material
models are required. The simplicity of
tensile testing has facilitated the common
practice of fitting tensile data to a
material model for use in an FE simulation
aimed at predicting sheet strain
fields and potential failure locations in
the formed part. In many processes,
tensile deformation is not the dominant
mode of plastic deformation.4 A clear
demonstration of this may be found in
forming limit diagrams that compare
major and minor strains.5 The time-honored
approach of using material
models constructed from tensile test
data in FE simulations where metal
sheet undergoes biaxial deformation
was recently challenged by Taleff et
al.6 As metal alloys continue to evolve,
forming process designs will rely more
heavily upon FE simulations, spurring
demand for more accurate constitutive
models and experimental data.
Not surprisingly, there has been a
steadily-growing interest in biaxial
testing of metallic cruciform geometries
(i.e., two-dimensional analogs
of the uniaxial tensile geometry). The
arms or “spokes” of the cruciform,
which are aligned at 90°, intersect
over a planar gauge area. Load is
transferred through the gripper ends
of the spokes along two perpendicular
axes with a suitable test apparatus.
Strain fields in the gauge area can be
measured with circle grids which are
commonly applied in sheet formability
tests. The cruciform test is often
associated with providing biaxial
strains and hence it does not typically
explore enough strain paths that would
be sufficient to generate a forming
limit diagram.3 However, Yu et al.7 observed
that different stress conditions
can be acquired by varying the ratio
of the loads applied along the spokes,
thereby enabling study of complex
strain paths. A key issue is accurate
measurement of strain fields which
has been problematic. In general, cruciform
tests have proven to me most
useful for evaluating the work-hardening
behavior in biaxial tension and
determining anisotropic yield loci.8
Although there are standards for uniaxial
tensile testing,9 none exist for
biaxial testing of cruciform specimens
at the present time. In fact, there is no
“standard” cruciform geometry as evidenced
by the proliferation of geometries
in the literature. There are some
important factors that have led to this
state of affairs that is briefly reviewed
in this paper.
Commercial biaxial testing systems
are complex, expensive, and hence
relatively scarce; examples are detailed
in Samir et al.10 and Doudard et
al.11 A more common alternative is a
self-driven biaxial testing apparatus,
such as those developed at the Universite
de Sherbrooke,12 Tokyo University
of Agriculture and Technology,13
and the Institute of Metal Forming
Technology at Stuttgart University.14,15
Fixtures that transform uniaxial deformation
(typically within a uniaxial
load frame) into controlled biaxial deformation
are common.16–18 In spite of
this variety, the vast majority lack the
ability to deform metals above room
temperature. This limitation is significant since substantial deformation
can be obtained from lightweight alloys
such as titanium, aluminum, and
magnesium at elevated temperatures.
This has rendered superplastic forming
(SPF) and, more recently, a lower
temperature version of SPF known as
quick plastic forming (QPF),19 important
production processes. A case in
point is the common AZ31B wrought
magnesium alloy, which exhibits poor
room-temperature formability due to
basal slip. Non-basal systems become
activated at higher temperatures leading
to a dramatic improvement in ductility.
Warm stamping and gas pressure
forming processes therefore become
viable approaches to manufacturing
magnesium components.
Cruciform specimen design is of
comparable importance to testing instrumentation.
While a cruciform geometry
with a uniform thickness and
spoke width may be thought of as a
natural extension of a uniaxial specimen,
it is unsatisfactory for studies of
plastic deformation in the gauge area
due to stress concentration and strain
uniformity issues. This has stimulated
development of a variety of geometric
features. The first of several features
that we consider is shown in Figure
1a. Here, very thin slits are cut into the
spokes.13,20,21 Kawabara et al.13 reported
that the slits promote uniform strains
within the inner part of the gauge area,
irrespective of the loading ratio. A second
feature is a smooth notch geometry added to the four corners of the
gauge area, as shown in Figure 1b.22,23
The intention here is to promote higher
strains in the gauge area. Banabic et
al.14,15 used this specimen geometry to
construct ambient-temperature yield
loci of different materials. A third feature
is a reduction in thickness within
the gauge area to further localize deformation
there. An example is shown
in Figure 1c, which contains a central
“recessed” region (denoted by the solid
circle in the gauge area).18 Combinations
of the aforementioned features
have been proposed; examples of these
are shown in Figures 1d–f.8,24,25
Despite efforts that have produced
numerous cruciform geometries, we
have found that each suffers from one or
more of the following three drawbacks:
- Geometry is based solely on FE
simulation without experimental
validation7,18,26
- Geometry precludes substantial
plastic deformation in the
gauge area (initial yielding and
anisotropy are of primary interest)14,15,20,25,27
- Geometry is appropriate for ambient
temperature deformation (or
testing conditions preclude high
temperatures);12,14,15,20,25 those
targeting higher-than-ambient
temperatures are limited to small
plastic strains17,27,28
This article presents some new cruciform-
shaped specimen geometries
that are particularly well-suited for
elevated temperature biaxial testing
of lightweight materials to large plastic
strains. Several specimen designs
were proposed and tested using a custom
biaxial testing fixture capable of
stretching at multiple biaxial strain ratios.29 Viability of a specific cruciform
design was based upon the extent of
plastic deformation of its gauge area.
Strain fields in selected specimen
gauge areas were measured with digital
image correlation (DIC). See the
sidebar for experimental procedures.
| HOW WOULD YOU... |
…describe the overall significance
of this paper?
This article describes limitations
of existing cruciform geometries
for biaxial testing. Important
geometric parameters required to
achieve biaxial deformation in the
gauge areas of some new specimen
designs at elevated temperatures
are examined. Strain fi elds were
computed with digital image
correlation.
…describe this work to a
materials science and engineering
professional with no experience in
your technical specialty?
Cruciform testing, which is the
two-dimensional analog of uniaxial
tension testing, is commonly used to
evaluate work-hardening behavior
in biaxial tension and to determine
anisotropic yield loci.
…describe this work to a
layperson?
Metal specimens machined into
the geometry of a cross, commonly
referred to as cruciform specimens,
are simultaneously elongated along
perpendicular directions to study the
accumulation of plastic strains and
failure. |
CRUCIFORM GEOMETRIES AND MATERIALS
Two specimen “families,” designated
as TxRxxx or NxxxDxxx, were investigated.
The spokes of the TxRxxx
specimen geometries were tapered at
an angle denoted as Tx (in degrees),
with a smoothly varying thickness
profi le denoted by radius Rxxx (in
mm), as illustrated in Figure 2a. The
through-thickness taper was selected
such that the final thickness at the center
of the gauge area was around 1.0
mm. The NxxxDxxx specimens, illustrated
in 2b, have neither a side taper
nor a thickness taper. Rather, they have
corner notches with depth Nxxx (in
mm) and a circular (flat-bottomed) recess
in the gauge area of diameter
Dxxx (in mm). The depth of the recess
was selected such that the final thickness
at the center of the gauge area is
approximately 1.0 mm. Specimens
were prepared from AA5083, Mg
AZ31B-H24,31 and TWIP steel.32
Specimen outlines were machined
with water jet cutting using a fine
abrasive and slow feed rate to guarantee
superior edge quality; all edges
were lightly polished subsequent to
machining. Thickness tapers resulted
from high speed milling. Designations
and dimensions for all specimen geometries
are listed in Tables I and II,
with examples in Figures 2 and 3.
Specimen designs were contingent
upon the initial gauge of the as-received
materials.
Each specimen was cut such that its
four spokes were oriented at 45° relative
to the rolling direction so as not to
bias deformation toward one of the
pulling axes. A circle grid pattern (2.5 mm in diameter circles) was electrochemically
etched in the gauge areas
of several specimens, as shown in Figure
3a,b.
Gripping was followed by a 15 minute
heating period. Before testing, the
grips were tightened to the point where
the slightest axial force in the load
cells was sensed. A test was then started
at a fixed rate of 0.05 mm/s, and
maintained until a preset deformation
was achieved, at which point the specimen
was removed from the fixture.
Lastly, for each specimen geometry,
several test specimens were deformed
to progressively larger limits (in 0.25
or 0.5 mm increments), up to fracture.
All specimens were tested without
prior annealing.
TESTING INSIGHTS
Non-viable Geometries
Visual evaluation of deformation
and failure was initially used to judge
the extent of plastic deformation in the
specimen gauge areas. For the TxRxxx
specimens, improved deformation localization
resulted when going from
the largest to the smallest R values in
Table I. Strain accumulation in the geometries
with the largest R values
(T4R525, T0R286, T1R286, T4R286)
occurred (primarily) outside of the
gauge area for all three materials. This
is depicted in Figure 4a–c, which show
fracture in three different geometries.
For example, the grid circles of
T4R525 in Figure 4a suggest essentially
no deformation in the gauge
area; strains accumulated outside of
the gauge area. This is further demonstrated
in Figure 5a–c which shows
test results for T4R286 Mg AZ31B
specimens at cross-head displacements
of 1.5, 2.5, and 3.5 mm, respectively. Biaxial deformation in the
gauge areas of these specimens was
therefore not possible. Ancillary tests
led to similar conclusions for the flat
cruciform specimens (R = ∞). Thickness
tapering was intended to shift the
deformation into the gauge area; however, the values chosen for the specimens
in Figures 4 and 5 proved to be
insuffi cient. Increasing T helped to
shift plastic deformation closer to the
gauge area, yet not into it. It was therefore
concluded that values of R smaller
than those chosen for the specimens
in Figures 4 and 5 should be examined.
The specimen geometries in Figures
4 and 5 were eliminated from further
consideration.
As for the NxxxDxxx family, the
two geometries without a circular recess,
N337D000 and N225D000,
proved to be non-viable. Specimens
with the geometry in Figure 4d always
failed along the intersection between
one of the spokes and the gauge area
due the corner notch. The absence of a
thickness recess (D for these two geometries
is zero) prevented deformation
from moving further toward the center
of the gauge area. In effect, N is
similar in function to T in that it promotes
plastic deformation closer to
the gauge area, but not within it.
Viable Geometries
Three cruciform geometries were
deemed viable when signifi cant deformation
of their gauge areas was observed.
These are T1R051, T4R051,
and N225D635. Figure 6a–c shows
the progression of plastic deformation
within the gauge area of T1R051 Mg
AZ31B specimens, at 1.5, 1.75, and
2.0 mm displacements, respectively. A
distinctive surface roughening of the
gauge area that is especially prominent
in Figure 6b is noted. Deformation
in the gauge area localizes by
means of two slowly evolving diagonally
oriented shear bands that intersect
the corners of the gauge area.
These are indicative of biaxial deformation
in the gauge area. Although
the bands grow at nearly the same rate
during deformation, fracture along
one of them ultimately occurs, as
shown in Figure 6c. Figure 7 shows a
similar progression for the T4R051
geometry. Deformation in the four
spokes of the tests detailed in Figures
6 and 7 is substantially less than that
noted in the other specimen geometries
in Table I.
The apparent success of T1R051
and T4R051 for Mg AZ31B suggests
the possibility of a fundamental relationship
between T and R that may be
material independent. Further evaluation
of the relative impact of T and R
will be required with additional testing
of other materials (e.g., AA5083)
aided by finite element simulation.
Some comments regarding the orientation
of fracture in the gauge area
relative to the rolling direction are
warranted. Figure 8a and b shows additional
test results for T1R051 and
T4R051 Mg AZ31B specimens, respectively.
The red arrow in each image
indicates the rolling direction. In
these and all other tests of the T1R051
and T4R051 geometries, fracture
eventually occurred along that shear
band with a perpendicular orientation
relative to the rolling direction. Interestingly,
the T4R051 showed a greater
tendency for this behavior at earlier
testing stages than the T1R051. The
consistency of the results in Figure 8
eliminates the possibility of a fixture-induced
departure from balanced-biaxial
deformation. While ongoing efforts
are focused on linking texture to
fracture, we note that the CRSS values
of basal and prismatic slip are lower
than that of twinning near 300°C.33
Fracture surfaces follow a 45° angle
relative to the specimen thickness.
STRAIN FIELDS
Measurement of strains from circle
grids on sheet surfaces suffers from a
number of well-documented drawbacks.3 Digital image correlation
(DIC), a fast and precise method for
measuring in-plane deformation and
displacement fields of plastically deforming
sheet materials, is a less problematic
alternative.32,34–37 The input to
the correlation algorithm requires a set
of digital images that store deformation
history from one surface of the
deforming sheet. Once a test is completed,
digital grids are computed on
each with a DIC algorithm. Displacement
and strain fields are computed
from the grids in a cumulative manner
through comparison of the first recorded
image with each subsequent
image. The spatial extent of the grid is
user-chosen and the region of strain
measurement is not restricted as is the
case when strain gauges or extensometers
are used (see Figure 1a,d,f). Digital
image correlation analysis was
performed with the SDMAP3D software38
which employs the correlation
algorithm of Sutton et al.39
The images in the present study
were computed from grids with a 10
pixel spacing and a 60×60 pixel subset
(0.72 mm × 0.72 mm) using a 5×5
pixel (local) strain gauge. A Canon
Powershot SX10 IS camera (1 f/s) was
used to record 1500×1500 bitmap images
(each 6.6 MB) during testing.
The pixel spatial resolution was 0.03
mm/pixel. Note that superior image
clarity was guaranteed by the camera
lens which had a large depth of focus.
A sapphire plate was placed between
the specimen surface and the digital
camera to protect the camera from excess
heat. Each specimen was lightly
polished prior to application of a grayscale
contrast pattern consisting of
white and black spray paint (high temperature)
droplets.
The white box superimposed on the
gauge area of a N225D635 Mg AZ31B
specimen in Figure 9a denotes the 8
mm × 8 mm DIC region of interest.
Digital grids computed over this area
provided true effective strain contours
at selected testing stages in Figure
9b–d (displacements increase from b
to d as do the peak strains). Strains
concentrate in a nearly square region
centered about the gauge area with
bands at effective strains of 0.5 intersecting
the corners of the gauge area.
This is denoted by the green contours
in Figure 9c,d. Local peak strains regions
in Figure 9c–e (see the yellow
and red contours in each) denote
growth of the dominant shear band
along which fracture of the specimen
ultimately occurred. This band (not
shown) was also oriented at 90° to the
rolling direction.
Figure 10 is another set of DIC
strain maps computed for a T1R051
Mg AZ31B specimen using a region
of interest that was identical to that in
Figure 9a. Plastic strains accumulate
near the center of the gauge area. Note
that peak strain levels reach 0.26 at the
upper left corner of the map in Figure
9d and decrease to 0.23 near the center
of the gauge area. Again, fracture occurred
along a single shear band as
noted by an increase in the strains associated
with these contours. Strain
levels were lower in the T1R051 specimen
relative to those of the N225D635
due to the geometry. Similar observations
apply for the T4R051 specimens,
although formation of the dominant
shear band occurred at lower effective
strains than those shown in Figure 9
(i.e., earlier in the test). The strain
contour maps in Figures 9 and 10
demonstrate that plastic strains accumulate
in the gauge areas of the
N225D635 and T1R051 geometries
and that deformation is biaxial up to
formation of the dominant shear
band.
CONCLUSIONS
Biaxial deformation of AA5083,
Mg AZ31B, and TWIP steel cruciform
specimen geometries was investigated
with a custom testing fixture at 300?C.
Combinations of spoke taper angle
and thickness taper that promote plastic
deformation in the gauge area were
determined. Both visual observation
of the tested specimens and strain
mapping via digital image correlation
revealed the extent of plastic deformation
in the gauge areas. In those cases
where biaxial deformation occurred
within the gauge area, crossing shear
bands developed with strains accumulating
along one of the two bands up to
fracture. Current experiments are focused
on understanding the role of
texture and other microstructural
mechanisms on biaxial deformation in
Mg AZ31B.
ACKNOWLEDGEMENTS
The assistance of P.D. Zavattieri
with computer programming aspects
of this research is gratefully acknowledged.
S. Agnew, R. Verma, A. Bower,
and E. Taleff generously shared their
knowledge of Mg AZ31 metallurgy and
forming. Cruciform specimens were
designed in Unigraphics by the GM
R&D design group.
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F. Abu-Farha is an assistant professor in the Department
of Mechanical Engineering, Penn State
Erie, Erie, PA 16563; L.G. Hector, Jr., is a staff research
scientist with the GM R&D Center, Warren,
MI; and M. Khraisheh is a professor in the Center
for Manufacturing, University of Kentucky, Lexington,
KY, and also with the Masdar Institute of
Science and Technology, Abu Dhabi, United Arab
Emirates.
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