Diffuse Bifurcation and Strain Localization in Alporas
Aluminum Foam
Dominick J. Werther1 and Dr. Kathleen A. Issen2
Department of Mechanical and Aeronautical Engineering
Purpose:
Aluminum foam is a relatively new material and its full range of engineering applications is not
yet known. One reason for this is that its material properties have yet to be described in such a way that
enables mathematical modeling of the material under different loading conditions. Once aluminum foam
is accurately characterized in terms of Young’s modulus, yield strength, and plateau stress, as well as cell
morphology and cell defects, finite element analysis should be able to accurately predict how the material
will behave in various applications. This will allow engineers to determine what applications the material
is suitable for as well as allow them to incorporate safety factors into mechanical calculations.
In the preliminary studies that have been done at Clarkson University on Alporas aluminum foam,
two things have been noted. Prior to the first peak of the stress strain curve, both a diffuse bifurcation,
which can be compared to buckling, and strain localization in the form of compactions bands are
occurring. Additionally, most of the strain localization in Alporas specimens is occurring near the ends of
the specimen, either top or bottom, which makes it difficult to see the bulk material reaction. My research
will examine various aspect ratios, as well as compare epoxied vs. non-epoxied ends, to see if the diffuse
bifurcation and strain localization can each be achieved individually as the only mode of deformation.
Another goal is to force strain localization to occur away from the ends of the material to see how the
bulk material behaves, rather than strain localization at the ends, which is likely due to end effects.
Background:
Aluminum foam is manufactured using a few different methods. The foam which will be tested
in this study, trade named Alporas, is made by first mixing calcium into the molten aluminum to increase
viscosity, then introducing TiH2 which separates into titanium and hydrogen. The titanium mixes with
the aluminum alloy and the hydrogen causes foaming of the molten mixture which is responsible for
creating the porous aluminum foam (Akiyama, et al. 1987).
The deformation of cellular metal foams has been characterized into five stages, which apply to
five distinct regions on the stress-strain curve. First is a possibly elastic linear section. A plastic near1
2
Class of 2005, Department of Mechanical and Aeronautical Engineering, Clarkson University, Honors Program
Project Mentor, Department of Mechanical and Aeronautical Engineering, Clarkson University
linear section follows. Then as the non-linearity sets in, one or more compaction bands are formed. After
the peak stress is reached an “oscillating” plateau occurs, where small but distinguishable peaks and
valleys are recognized. These stress drops are thought to match up with the creation of individual bands
as one weakens and accumulates material. The peaks are where the material is becoming denser before
the next band collapses. Then once the bands have all thickened, an overall densification of the material
is observed, followed by a stress increase as the material becomes vastly more dense, comparable to solid
metal (Lim, et al. 2002, Papka and Kyriakides, 1998, Sugimura, et al. 1997).
Studies have been done on bone to determine how the end conditions of platen testing play a role
in the overall behavior of the material undergoing compression. In the bone research, brass end caps have
been used to constrain the top and bottom so that the deformation is forced to occur in the middle, in a
“gauge” section (Keavney, et al. 1997). There have also been numerous studies done on how individual
cell deformation effects the overall deformation of the specimen. These studies have shown that
weaknesses in cell structure are a critical component in the overall collapse of the specimen. Such things
as curves, wiggles and cracks have been cited as weaknesses that can potentially lead to cell failure,
which in turn propagates to band collapse (Bart-Smith, et al. 1998, Bastawros, et al. 2000, Markaki and
Clyne, 2001, McCullough, et al. 1999, Sugimura, et al. 1997).
Aluminum foam has been studied at Clarkson University by Dr. Kathleen Issen and her students.
There have been two types of foam studied: Alporas and Cymat. The Cymat has been studied more
extensively and has a great propensity for forming one compaction band and then propagation (thickening)
of that one band. The Alporas, although less studied, has more uniform cell morphology. It is proposed
that this more uniform structure is responsible for the densification at the ends of the specimen rather than
in the middle. The Cymat compaction band almost always forms in the middle, but that may be due to
the ends being more dense than the middle because of the manufacturing process by which Cymat
aluminum foam is created.
Proposed Work:
Although there has been considerable research done on Alporas aluminum foam and its behavior
during deformation, there is no mention in the literature of a diffuse bifurcation or buckling mode. In the
Alporas specimens that have been tested at Clarkson University thus far, this buckling has occurred fairly
regularly, pre-peak on the stress-strain curve. The buckling has been identified using both the naked eye
as well as digital image correlation, which is capable of showing the horizontal displacements of the
specimen during its compression. Hypothetically, changing the aspect ratio will isolate this behavior. It
is expected that short specimens with large surface area should not buckle, whereas long specimens with
small relative surface area should buckle more readily.
In the preliminary specimens tested, the majority have shown strain localization at the top platen.
This is possibly due to the weaker half cells that are on all sides of the specimen. Since these half cells
are weaker than the full closed cell structure, compaction bands may form near the top due to cell
instabilities there. Hypothetically in this case, potting the ends with epoxy will eliminate the half cell
instabilities forcing the localization to occur toward the middle of the specimen.
I plan to study the material behavior of Alporas aluminum foam during compression, with
specific attention to aspect ratio and end treatment. I will design a statistical model to test three variables.
First is how the specimen was cut: fly-cut vs. wire electrodischarge machined. Second will be end
treatment. Some specimens will remain as is while others will have the ends potted in epoxy. The third
variable will be the aspect ratio, or ratio of a height to base length of the specimen. Specimens will be
tested with an intermediate ASTM standard compression test aspect ratio, 2:1, and a small aspect ratio,
1.5:1.
The faces of the material that will be studied must be painted flat black and then speckle painted
with white to create a unique pattern which can be correlated with the Vic-2D image correlation software.
Digital images will be taken of two faces of the material during the compression. These images, coupled
with force and displacement data, will be converted into force vs. image number and stress vs. strain plots.
Vic-2D will use the camera images as inputs, and the outputs will be surface strain maps of two of the
specimen’s four faces.
The end product will be a useful comparison of the results of the different test variables. If the
results prove to be conclusive, regarding the ways in which the foam should be tested in order to constrain
for a certain deformation behavior, this information could prove to be very useful to the research
community, as well as industry, in creating a standard for the compression testing of the material.
Preliminary Results:
When testing the Alporas specimens, originally specimens were compressed 20mm in order to
see the stress-strain behavior of the material. The number of oscillations in the stress plateau should be
indicative of either the number of collapse events or buckling events taking place.
Initial specimens were tested with a 2:1 aspect ratio and no treatment of the ends. In certain
specimens the diffuse bifurcation or buckling can be seen with the naked eye. This result is validated by
correlations of these specimens which show u-displacements indicative of such an event. For example the
top of the specimen moves left and the bottom moves to the right. Correlations have also shown
consistent strain localization at the top of the specimen.
In a test specimen with a smaller aspect ratio, an expected Poisson’s effect in the u-displacement
correlation was observed (lateral fattening with axial shortening), which indicates that no buckling
occurred. Additionally, two discrete bands formed in the middle of the specimen. This is a promising
preliminary test in terms of being able to isolate strain localization from diffuse bifurcation.
When testing a specimen with a 2:1 aspect ratio and epoxied ends, a compaction band formed
below the epoxied end. This is promising because the goal of filling in the half-cells was to force the
compaction band to form in the middle of the specimen and not at the ends. This is what we have
achieved with this test.
Summary:
Tests have been run to determine the behavior of Alporas aluminum foam under uniaxial
compression. Two problems encountered were strain localization at the ends and diffuse bifurcation of
the specimen. I hope to test end treatment and aspect ratio variations to see if these deformation modes
can be separated and controlled. The tests I have run this summer have shown that the hypothesis I have
developed may actually be true and pertinent to the research community studying aluminum foam. Many
more tests must be run to statistically validate the preliminary results I have observed.
References:
Akiyama, S., Ueno, H., Imagawa, K., Kitahara, A., Nagata, S., Morimoto, K., Nishikawa, T., and Itoh, M., 1987,
“Foamed metal and method of producing same,” in US Patent 4,713,277.
Bart-Smith, H., Bastawros, A-F., Mumm, D.R., Evans A.G., Sypeck, D.J., Wadley, H.N.G., 1998, “Compressive
deformation and yielding mechanisms in cellular Al alloys determined using X-ray tomography and surface strain
mapping,” Acta Mater., 46, pp. 3583-3592.
Bastawros, A.F., Bart-Smith, H., Evans, A.G., 2000, “Experimental analysis of deformation mechanisms in a
closed-cell aluminum alloy foam,” J. Mech. Phys. Solids, 48, pp. 301-322.
Keaveny, T.M., Pinilla, T.P., Crawford, R.P., Kopperdahl, D.L., Lou, A., 1997, “Systematic and random errors in
compression testing of trabecular bone,” J. Ortho. Research, 15, pp. 101-110
Lim, T.-J., Smith, B., McDowell, D.L., 2002, “Behavior of a random hollow sphere metal foam,” Acta Mater., 50,
pp. 2867-2879.
Markaki, A.E., and Clyne, T.W., 2001, “The effect of cell wall microstructure on the deformation and facture of
aluminum-based foams,” Acta Mater., 49, pp. 1677-1686.
McCullough, K.Y.G., Fleck, N.A., Ashby, M.F., 1999, “Uniaxial stress-strain behaviour of aluminum alloy foams,”
Acta Mater., 47, pp. 2323-2330.
Papka, S.D., and Kyriakides, S., 1998, “Experiments and full-scale numerical simulations of in-plane crushing of a
honeycomb,” Acta Mater., 46, pp. 2765-2776.
Sugimura, Y., Meyer, J., He, M.Y., Bart-Smith, H., Grenestedt, J., Evans, A.G., 1997, “On the mechanical
performance of closed cell foams,” Acta Mater., 45, pp. 5245-5259.