Compaction Band Formation and Propagation Behavior in
Closed Cell Aluminum Foam
David Brush ’05, Mechanical Engineering Department
Faculty Advisor: Kathleen Issen, Mechanical Engineering Department
Abstract:
Aluminum foam is a porous, lightweight alternative to solid aluminum which is
relatively new to the market. As such, not much is known about the material, despite its
possible applications in sound and vibration damping devices, a lightweight
constructions, and impact energy absorbers. Further understanding of the material, and
specifically its tendency to form regions of high local strain during compressive
deformation, is the goal of this research. Similar research performed by William A.
Olsson on Sandstone will be used in part as a guideline, and an attempt to adapt his
sandstone theory to aluminum foam will be made.
Background:
Compaction bands, or strain
C5a15 Stress Strain Curve
8
localizations, are defined as regions
7
of high local strain within a
bands are an apparently common
5
Stress (MPa)
material being compressed. These
6
4
3
failure mode among highly porous
solids, including aluminum foams
(Bastawros et al. 2000; Bart-Smith
2
1
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Axial Strain (Crosshead)
et al. 1998), sandstone (Olsson
2001), Steel Foam (Park & Nutt
Figure 1: Stress-strain plot (with unloading loops) of the foam sample
referenced in this paper. Note the stress drop and subsequent plateau
region.
2001), and polycarbonate honeycombs (Papka & Kyriakides 1998). The formation of
these bands occurs in two stages. In the first stage, band formation, a single void, or cell,
fails, due to load concentration and/or weak cell geometry. The collapse of this cell leads
to the collapse of surrounding cells, thus causing the failure to propagate across the
material until a complete layer of crushed cells forms (Daxner et al. 1999). During this
failure, there is unloading of the surrounding cells in the material, and therefore a drop in
the applied load, corresponding to an overall stress drop in the material. This
phenomenon can be thought of as a series of three springs. Up until the beginning of
band formation, these springs all have a constant elastic modulus, and act as a single
spring. However, when the band forms, the middle spring’s modulus gradually
decreases, thereby causing the middle spring to take up some of the displacement of the
surrounding springs. The second, longer stage (band propagation) is when this original
band propagates not across the material, but along it (parallel to the loading). Not all
materials that form bands will exhibit propagation, however, many form multiple discrete
bands instead. No matter whether a material propagates or forms multiple bands, this
stage is characterized by a largely constant stress within the material, indicated by a
“plateau region” on stress-strain plots.
Similar research has been performed regarding the localization behavior of
sandstone regarding propagation behavior (Olsson 2001). Linking of his results and
those gathered from this research would make excellent progress toward a unified theory
for compaction band formation in many, if not all porous solids. Of particular interest is
the equation relating band propagation speed and overall deformation rate to change in
porosity
Where vr and vp are the rate of band thickening and overall specimen shortening,
respectively, and P and Po are the band and original material porosities. The relation
between relative propagation rate and energy absorption is also of interest, and is given
by
Where E – Eo is the change in strain energy, and σ is the plateau stress of the
material. This second relation is of special interest due to the great potential aluminum
foam has as an impact energy absorption material.
Understanding this failure mechanism better is important for bringing porous
materials into widespread use. Porous metals, such as aluminum foam, are already
known to have many uses, including lightweight construction/filler material, sound
damping, heat exchange, and impact energy absorbers (Sugimura et al. 1997; Bastawros
et al. 1998; Daxner et al. 1999; Park & Nutt 2001). In construction materials, it is
imperative that the material not be loaded past its peak stress, as doing this would lead to
a massive compressive failure. Compression fractures in persons with osteoporosis may
be looked at as such a failure. Also, as foam metals do not exhibit a fully elastic initial
deformation, pre-loading of structural foam may be desired. Impact energy absorption
applications, however, make full use of this same phenomenon, the plateau region of
foam failure providing excellent energy absorption over a large strain at a nearly constant
stress.
Current/Planned Work:
The focus of this research is on the band
formation and propagation behavior in Aluminum
foam, in particular Cymat. Samples of the material
have already been compacted in a displacementcontrolled load frame, and images of one face of each
specimen were taken throughout the process. Thus, the
data for this research has already been gathered,
although if the need arises, more samples can be tested.
Figure 2: Full color strain map. The line
down the center is the location of the vertical
slice taken to and graphed in figure 3.
Analysis starts in Vic2D. This program takes
digital images and correlates them with a base image, finding displacements of the
material imaged, then using those displacements to define local strains. Base images
must be chosen based on what region of specimen deformation is to be analyzed, since
too much difference between the two images analyzed will lead to a correlation
breakdown and unusable results.
Some preliminary analysis of the stress drop region has already been performed.
1 dimensional “slices” of strain data were taken from a sample both halfway down the
stress drop and at the base of the stress drop. Numerical integration reveals that at both
points along the drop, 5% of the total displacement put into the band (defining the band
region as everywhere compressive deformation occurred) came not from the moving load
frame, but from the unloading of the surrounding material. Since the material exhibits
linear elastic unloading, this seems to indicate that there is a linear correlation between
the drop in stress during band formation and the displacement put into the band.
One basic question that must be answered in the course of this research is how to
define a strain to use as band strain. For the above calculations, all negative strains are
considered to be in the band. That same preliminary data, however, shows that some
material far outside of the band region also exhibits some negative strain. One possible
answer is to look at either the maximum or average unloading strain, and use some
percentage of that value as the boundary value for the band region. Another possibility is
to use the overall specimen strain, either at every individual point, or perhaps at the onset
of band formation. At first, a logical choice for this boundary would seem to be at 0
strain, however this approach would only yield reasonable results when unloading outside
of the band occurs (I.E. during band formation).
Analysis of the stress drop
Strains down central pixels (x=493)
will be done by correlating digital
0.01
0
0
100
200
300
400
500
600
700
800
900
1000
images taken of already-crushed
specimens taken during the drop
-0.02
in Vic2D. Initially, the 2-
strain
-0.01
-0.03
dimensional array of local strains
-0.04
produced will be analyzed by
-0.05
isolating “strips” of data going
-0.06
Y-axis location
Figure 3: Vertical strip plot of the region noted in Figure 2. The large dip is
the strain localization region, and is negative (compressive). The region
outside of this exhibits a slight positive strain (unloading).
through the specimen, holding
either the horizontal or vertical
coordinate constant. These strips
can then be used to create plots which correlate position and strain. Horizontal strips will
be taken to look for edge effects, or more accurately corner effects, in the material,
mainly to ensure that data is not taken too close to the sides of the specimen throughout
the rest of the analysis. Vertical strips will be used to examine the relation between the
unloading region and band region of the specimen during the stress drop, both to look for
general trends and to obtain maximum and average unloading strains if one of these is
chosen as a candidate for the definition of the boundary between band and non band
strain. Also, band thickness at the bottom of the stress drop will be found, and compared
with the size of the stress drop for many different samples in an attempt to correlate the
two phenomena (It is expected that larger stress drops will correlate with thicker initial
bands).
In the plateau region, analysis will be done using
complete strain maps. Data will
first be loaded into TecPlot, where it can be more easily
manipulated. Using the definition for band strain chosen
from those discussed, the strain maps will be reduced to
black and white maps, with all locations in the sample
with local strain exceeding the definition of band strain
will be black, and everything else white. By using a
pixel counter (Photoshop), the area of this black region
can be quickly determined, and by dividing this area by
the image width, an average band thickness is
determined. This band thickness, calculated at regular
intervals within the plateau region of deformation, will be
Figure 4: Two-tone strain map used to
determine average band thickness.
used to try to compare the behavior of the foam to current
sandstone band propagation theory which has been formulated by Olsson.
Schedule of work to be done:
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Summer 2004: Most of the outlined data processing, initial data analysis,
possible compaction of more specimens
Weeks 1-2: Stress Drop Correlations taken, horizontal cross sections taken and
analyzed for corner effects.
Weeks 3-4: Vertical cross sections taken, analyzed for trends.
Weeks 5-6: Stress Plateau Correlations taken, comparison of different definitions
of minimum band strain made to determine best definition.
Weeks 7-9: Strain maps converted to black and white band/no band images,
calculation of average band widths made.
Week 10: Attempt to formulate equations governing band propagation, possibly
by adjusting Olsson’s results.
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Fall 2004:
September-October: Completion of data analysis, compaction of additional
specimens (if needed) continued work on adapting Olsson results to foam,
continued analysis of Stress Drop data.
November-December: Writing of final paper.
Possibilities for Future Research:
As noted previously, not all materials which form compaction bands form only
one band which then propagates. Many materials, such as sandstone and other brands of
aluminum foam such as Alporas, form multiple discrete bands. The reason for this is yet
to be determined however. Additionally, many materials, such as sandstone, have a
much smaller stress drop at band formation, but again,
the reason for this is unknown, and may even be related to the number of bands formed,
since Alporas and sandstone have much smaller stress drops than Cymat, and both of
them also exhibit much smaller stress drops. Also, it might be worthwhile to compare
stress drop size and band width to the ratio of cell size to specimen height, as it seems
logical that all of these would be somehow related to one another. Finally, porous
materials other than aluminum foam and sandstone exhibit similar localization and
propagation behavior, and would benefit from similar analysis to this research.
References:
Bart-Smith H, Bastawros AF, Mumm DR, Evans AG, Sypeck DJ, Wadley HNG,
Compressive deformation and yielding mechanisms in cellular Al alloys
determined using X-ray tomography and surface strain mapping, Acta Mater, 46
(10), 3583-3592, 1998.
Bastawaros AF, Bart-Smith H, Evans AG, Experimental analysis of deformation
Mechanisms in a closed-cell aluminum alloy foam, Journal of the Mechanics
And Physics of Solids, 48, 301-322, 2000.
Daxner T, Bőhm HJ, Rammerstorfer FG, Mesoscopic simulation of inhomogeneous
metallic foams with respect to energy absorption, Computational Materials
Science, 16, 61-69, 1999.
Olsson WA, Quasistatic propagation of compaction fronts in porous rock, Mechanics of
Materials, 33, 659-668, 2001.
Papka SD, Kyriakides S, In-plane crushing of a Polycarbonate Honeycomb, Int. J. Solids
Structures, 35 (3-4), 239-267, 1998.
Park C, Nutt SR, Anisotropy and strain localization in steel foam, Materials Science and
Engineering, A299, 68-74, 2001.
Sugimura Y, Meyer J, He MY, Bart-Smith H, Grenstedt J, Evans AG, On the mechanical
performance of closed cell Al alloy foams, Acta Mater, 45 (12), 5245-5259, 1997.