Compaction Band Formation and Propagation
Behavior in Closed Cell Aluminum Foam
David Brush1 and Kathleen Issen2
Department of Mechanical and Aeronautical Engineering
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 light weight construction, 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.
Compaction bands, or strain
C5a15 Stress Strain Curve
8
localizations, are defined as regions of high
7
local strain within a material being
6
compressed. These bands are an apparently
5 (MPa)
Stress
common failure mode among highly porous
4
3
solids, including aluminum foams (Bastawros
2
et al. 2000; Bart-Smith et al. 1998), sandstone
1
(Olsson 2001), Steel Foam (Park & Nutt 2001),
0
0
0.005
0.01
0.015
0.02
0.025
0.03
Axial Strain (Crosshead)
0.035
0.04
0.045
0.05
and polycarbonate honeycombs (Papka &
Figure 1: Stress-strain plot (with unloading loops) of the foam sample
Kyriakides 1998). The formation of these
referenced in this paper. Note the stress drop and subsequent plateau
bands occurs in two stages. In the first stage,
region.
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
1
2
Class of 2005, Mechanical Engineering, Clarkson University, Honors Program; Oral Presentation
Associate Professor, MAE Department, Clarkson University
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take up some of the displacement of the surrounding springs, and also lessening the load required to keep
the springs compressed. 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.
The focus of research thus far has been to analyze the behavior of aluminum foam down the
stress drop, when the band first forms. This was done using surface images of aluminum foam samples
compacted in the course of previous experimentation. By processing these images using an image
correlation software package, Vic2D, a 2 dimensional “map” of surface strain can be produced (strain
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being calculated from one image to another)
One very interesting trend appears when looking at these local strain maps. By taking a one
dimensional “slice” of data coincident with the loading of the specimen, one can get a sort of band strain
profile, as seen below. The data points on such a profile, when shifted slightly, correspond extremely
well to an exponential plot defined by the equation
Where A is the maximum local strain
Strains down central pixels (x=493)
across the band region, and B is a constant
0.01
0
0
100
200
300
400
500
600
700
800
900
1000
which scales the width of the plot. B does not
appear to change much, if at all, throughout the
-0.02
stress drop, indicating that the width of the
-0.03
band does not change at all throughout the
-0.04
drop, indicating a formation event with little or
-0.05
no propagation, as was expected. This curve,
strain
-0.01
interestingly enough, is in the same family of
-0.06
Y-axis location
curves as the bell probability curve. Taking
two derivatives of this relation, we then find an expression for the value of x at inflection, and plugging
this value back into the equation, we find that the inflection point of this curve is located at 37 percent of
the maximum strain. Since inflection in this plot represents where the local strain increases the fastest,
this point gives us a good approximation of where the band “front” would be located. If this phenomenon
were ideal, the strain would instantaneously jump from a low, non-localized, to a much higher band strain
(think of the spring example from above, each of the springs would have its own constant strain
throughout it, but the middle spring’s strain would be much higher due to a lower elastic modulus). Thus,
at the band front, an infinite slope would be observed in the strain plot.
In research to come, this definition of band strain will be used to analyze the propagation of the
compaction band beyond its initial formation, and comparison of the behavior of aluminum foam and
Olsson’s sandstone will be made.
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
85
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.
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