HIGH STRAIN RATE RELOADING COMPRESSION TESTING OF A CLOSEDCELL ALUMINUM FOAM
A. Taşdemircia, M. Güdena, I. W. Hallb
a
Department of Mechanical Engineering, Izmir Institute of Technology Gülbahçe Köyü, Urla, Izmir,
Turkey 35430
b
Department of Mechanical Engineering, University of Delaware
a
Newark, DE 19716, USA
ABSTRACT
3
Al-closed-cell foam with densities varying between 0.35 and 0.50 g/cm was reloaded in SHPB compression testing in order to
determine the flow stresses at large strains at almost a constant strain rate. The sample reloaded six times between the bars
of the SHPB and the subsequent stress-strain and strain rate behaviors of the sample were calculated. The results were
compared with quasi-static testing. The stress-equilibrium during SHPB testing was determined through the numerical
modeling of the testing. The foam showed almost no strain rate dependency of flow stresses, while a marked effect of foam
density was found on the flow stress values. It was shown numerically that in high strain rate testing the variation of the stress
equilibrium decreases in the subsequent loading cycles as the foam density increases.
Introduction
Closed cell aluminum foams have received much attention in recent years as energy absorbing materials for their ability to
undergo extensive deformation at relatively low stresses (plateau stress). Potential applications of metal foams include light
weight cores for sandwich panels, shells and tubes where the foam can increase the resistance to local buckling, increase the
impact resistance, and improve the energy absorbing capacity of the structure [1-3]. Al foams are also of possible interest for
ballistic applications because they present a very large acoustic impedance mismatch with common armor materials, offering
the possibility of being able to modify the way in which elastic waves travel through multi-component armor [4].
Therefore two cases of energy absorption are of particular practical interest. The first concerns blast protection, in which
loading occurs over a broad area, and the second concerns resistance to penetration by small projectiles, in which severe
point loading is experienced. In both cases, the loading or strain rates are higher than those of static or quasi-static rates and
which may significantly change the mechanical response of the materials. Therefore, in designing with metallic foams with
energy absorbing parts, the mechanical properties are needed for strain rates corresponding to those created by impact
events. Quasi-static and high strain rate mechanical behavior of metallic foams has been fairly extensively studied and
reported, e.g. [5-9]. In the high strain rate testing of Al foams using compression type Split Hopkinson Pressure Bar (SHPB),
the strain attained is relatively small. To attain higher strains, the strain rate should be increased in which the mechanical
behavior of the foam sample may change. Also in regular SHPB testing, the setups having different bar lengths, it is normally
impossible to examine the whole stress-strain curve since while using the metallic rods with different lengths the greatest
obtainable displacement is fairly small compared with the levels needed to deform the foam material until its densification
strain. In order to avoid this limitation, reloading tests can be applied in the SHPB testing. In reloading tests, the strain rate
during multiple loadings almost remains constant as the sample deforms to larger strain levels in each successive loading. The
present work was initiated, in order to test relatively low density Al closed-cell foams in SHPB to attain large strain levels at a
fairly constant strain rate (strain rate decreases in subsequent loadings, but the change in strain rate between the two loading
was about 8%). The results of SHPB testing were compared with those of quasi-static testing. The deformation behavior of the
foam sample during SHPB testing was further modeled in order to confirm the validity of the reloading tests. The verified
material model constants may also be used for applications involving high strain loading of Al closed-cell foams.
Materials and Testing
Al closed-cell foam filler was prepared using the foaming of powder compacts (foamable precursors) process patented by
Fraunhofer CMAM [10]. The process started with the mixing of appropriate amounts of basic ingredients, Al and TiH2 (1 wt%)
powders, followed by an initial cold compaction inside a steel die of 7x7 cm in cross-section under a pressure of 200 MPa. The
average particle size of the Al powder was 34.64 μm and the size of TiH2 particles was less than 37 μm. The cold-pressed
compacts having 80% initial relative density were then open-die hot-forged at a temperature of 350oC under a pressure of 400
MPa, resulting in foamable precursors with the final densities of 98% and thicknesses of approximately 8 mm (Figure 1).
o
Foaming was performed in a pre-heated furnace at a temperature of 750 C and detailed information on the foaming process is
given in [11]. In order to prepare Al foams of different densities, the steel mold accommodating the foamed precursor was
taken from the furnace after various holding times and then water-quenched. Foam plates of 8x8 cm in cross-section and 3-4
cm in thickness (Figure 1) and having densities ranging between 0.35 and 0.5 g/cm3 were prepared. The weights of the drilled
cylindrical compression test specimens, 19.05 mm in diameter and 19.50 mm in length, were measured before compression
testing in order to calculate the relative densities. The X-ray radiography inspections of the cell structure of the foams had
clearly shown a dense Al foam structure in the regions next to foamed-plate skin. Further, a dense foam layer at the bottom of
the foam plate formed as a result of the liquid metal drainage during foaming at a high temperature. Since the foam plates
accommodated relatively homogenous cell structure across the normal to the foam expansion direction, the cylindrical foam
test samples for the compression testing were drilled normal to the thickness of the plates (normal to the foam expansion
direction) using electro-discharge machine. Drilled foam samples were washed with acetone and then dried in a furnace at
100oC for 2 h.
Figure 1. Images of the foamable precursor and foamed precursor, showing four times expansion in the initial thickness of the
precursor.
Quasi-static compression tests were performed in a Shimadzu AG-I testing machine, while the high strain rate tests were
conducted using a compression SHPB apparatus. The SHPB apparatus used consists of 19.05 mm diameter Aluminum 6061T6 bars, a 2700 mm long incident bar and transmitter bar, and a 356 mm long striker bar. Detailed information about the SHPB
used is given in [12]. The samples are compressed by accelerating the striker bar from a gas chamber so that it impacts the
incident bar. The resulting elastic wave travels down the incident bar to the specimen/bar interface where part of the wave is
reflected and part continues through the sample and into the transmitter bar. The incident, transmitted and reflected waves are
measured by strain gages on the bars. Among other factors, strain rate depends on the speed of striker bar generating the
•
elastic wave, and this is controlled by adjusting the pressure in the chamber. The strain rate ( ε ), the strain (ε) and the stress
(σ) of the tested sample were calculated using the following equations;
•
ε( t ) = −
2C b
ε r (t)
Ls
(1)
ε( t ) = −
2C b t
∫ ε r (t )
Ls 0
(2)
σ( t ) =
EbAb
ε t (t )
As
(3)
where, Cb is the elastic wave velocity in the bar, Ls is the sample length, As and Ab are the sample and bar cross-sectional
areas and t is the time, respectively. εr and εt are reflected and transmitted strains measured from strain gages on the bar,
respectively. The above equations are based on the assumption that the forces at sample-bar interfaces are equal. The
experimental force equilibrium for the same incident and transmitter bars diameter is expressed as
σ t (t ) = σi (t ) + σ r (t )
(4)
where, σt, σi and σr are the transmitted, incident and reflected stresses, respectively. The left and right sides of this equality
are used in the so-called "one-wave" and "two-wave" analyses, respectively. Stress equilibrium within the sample is
considered to be reached after 3-4 back and forth reflections of the wave in the sample and the time at which equilibrium is
established depends on the wave transit time of the sample [13].
SHPB experiments were performed in which six reflections of the waves were recorded. In reducing the data, no attempts
were made to account for dispersion or attenuation in the bar. Thus, for second and subsequent waves, the transmitted
(compressive) wave in the transmitter bar was simply subtracted from the previous compressive wave in the transmitter bar to
determine the net force on the sample. Then, the stress on the sample was calculated using Eqn. 3. The strain in the sample
in each loading was calculated using the Eqn.2. To determine the initial strain level of the second and subsequent loading, the
final strains attained in the previous loading was simply added.
The deformation process at high strain rates was observed by photographing the specimen sequentially in predetermined
short time intervals (of the order of few microseconds) using a high speed camera during SHPB testing. With the Ultra 8 highspeed camera used for the present study, a maximum of eight frames can be photographed at speeds up to 100 million frames
per second. In the time domain, the interframe time can be varied between 10 nsec and 1 msec. The camera can be
synchronized with the incident bar strain-gage or can be delayed to photograph the events of interest only.
Modeling
A three-dimensional SHPB finite element model was used to study the dynamic deformation behavior and stress-state of the
Al-foam specimens. The analyses were performed using the commercial explicit finite element code LS-DYNA 970. For each
test modeled, the force and, hence, the stress values were calculated at several locations within the sample, including the
sample front and back surfaces, as well as at the location of the strain gages on the incident and transmitter bars of the SHPB
apparatus. The results calculated from the model were compared with the output from the strain gages on the incident and
transmitter bars. The model has three components in contact; the incident and transmission bars (the lengths of which are
2700 mm), and the specimen. The bar and specimen diameters are 19.05 mm. Two axes of symmetry were assumed
therefore only one quarter of the bar was modeled as shown in Figure 2. The component materials were modeled with eight
nodes solid elements and the bar-specimen interfaces were modeled with the automatic contact sliding interfaces without
friction. A total of 75 elements were used in the model for the cross-section, which provided 10 elements across the radius of
the bars and a total of 400 elements were used along the length of the bar. Mesh biasing along the bar axis was utilized to
refine the meshes at the contact interfaces. Further details of the modeling procedure may be found elsewhere [13].
*MAT_HONEYCOMB (material 26) was chosen for the Al foam material model [14]. Material 26 offers uncoupled orthotropic
behavior as seen in foams. Nonlinear elastoplastic material behavior can be defined separately (for each direction) for all
normal and shear stresses. The aluminum bars were modeled with an isotropic elastic material model. Force equilibrium in the
specimen is further checked numerically using a dimensionless number, R [15],
2(F1 − F2 )
R=
(5)
(F1 + F2 )
where, F1 and F2 are the SHPB test sample front and back surface forces respectively. This number is a measure of the extent
of deviation from stress equilibrium in the specimen, which can further be compared with the experimental force equilibrium
given in Eqn. 4.
Figure 2. Finite element mesh used in the model of SHPB specimen (only specimen bar interface sections are shown)
Results
Figure 3 compares experimental results from SHPB tests with the corresponding numerical calculations. This figure presents
the stresses in the bars resulting from the incident, reflected and transmitted waves as a function of time for a foam sample
loaded six times (numbered in Figure 3). Despite the small differences in stress values between the experiment and the model
particularly at the beginning of the reflected wave and in the later part of the transmitted stresses, the shape of the model
stress-time curve and the maximum stress values show excellent agreement with those of the experiments. This indicates that
the model is indeed accurately capturing the mechanical property response of the samples to high strain rate deformation. The
variation of the strain rate in each loading numbered in Figure 3 is shown in Figure 4. The strain rate in each subsequent
-1
-1
loading decreases as shown in Figure 4. The average strain rate in the first loading is 820 s , while it decreases to 620 s .
The decrease in the strain rate between two subsequent loadings is less than 6%, but it is about 25% between first and last
loading.
Figure 5 shows the stress-strain curve of a foam sample of 0.38 g cm-3 foam sample tested in SHPB at strain rates varying
between 820 and 620 s-1 together with quasi-static strain rate stress-strain curve. The variation of the average strain rate in
each loading is also shown in the same figure. It is seen that the flow stress show the typical shape which may be divided into
three regions, 1) linear elastic region, 2) collapse region and finally, 3) a densification region in which the properties approach
those of fully-dense alloy. Also, strains in excess of 50% were easily achieved in SHPB testing. Despite to the small
differences between the flow stresses of quasi-static and high strain rates particularly in the initial collapse region of the stressstrain curves, the foam tested typically show almost no or very low strain rate sensitivity. The absence of marked strain rate
sensitivity is in agreement with other observations for this and other aluminum alloys [16-18]. It should also be noted that any
existed rate sensitivity in flow stresses could not be extracted because of unavoidable variations in the flow stress values of
the samples of the same density. The flow stress values of the foam are however seen to be a strong function of the density.
-3
Figure 6 shows the stress-strain curves of the foam samples of 0.38 and 0.42 g cm tested at very similar strain rates (8501
700 s ) in the SHPB. The foam sample of higher density seen in this figure shows higher flow stress than that of lower density
foam.
Bar stress (MPa)
150
reflected
100
50
0
1
2
3
4
-50
-100
incident
-150
0
transmitted
5
6
experimental
numerical
1000 2000 3000 4000 5000 6000 7000
Time (μs)
Figure 3. Experimental and numerical bar stress vs. time (Numerical data slightly time-shifted for clarity)
1000
-1
Strain Rate (s )
1
800
2
3
4
5
6
600
400
200
0
1000
2000
3000
4000
5000
6000
7000
Time (microsec.)
Figure 4. Numerical specimen strain rate vs. time
40
Stress (MPa)
35
30
25
-1
0.0011 s
20
15
10
-1
620 s
-1
820 s-1
770 s
-1
740 s
-1
660 s
700 s-1
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Strain
Figure 5. Compression stress-strain curves of Al foam at quasi-static and dynamic strain rates.
8
Stress (MPa)
7
6
0.38 gr/cm
3
0.42 gr/cm
3
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Strain
Figure 6. Comparison of stress strain curves of foams having different densities at similar strain rates
The numerical contact force history at the specimen-bar interfaces is used to calculate the dimensionless number, R, which
indicates the deviation from homogeneous stress state within the sample and is shown in Figure 7. The numerical prediction
shows that during the SHPB testing of foam, the value of R is quite large at the beginning and end of the foam deformation in
each loading as shown by circles in Figure 7. The variation of R values is however decreases during loading through the
subsequent loadings as the density of foam sample increases as the sample deforms to larger strains.
The sequence of deformation events was recorded during SHPB compression testing and several frames of a typical record
are presented in Figures 8 (a-d) as a function of strain. Figure 8(a) shows undeformed sample. As soon as compression
starts the thinnest sections of cell walls approximately parallel to the compression axis start to buckle and form a localized
deformed region as marked by arrow in Figure 8(b). In the later stages of deformation strains of 35%, in the collapse region
localized foam deformation is progressive and when the strain reaches strains of 60% (Figure 8(d) densification starts. The
above deformation mode of foam’s deformation is very much similar to the quasi-static strain rate deformation sequence.
R, Equilibrium parameter
1.0
0.5
0
-0.5
-1.0
0
1000
2000
3000
4000
5000
6000
7000
Time, (microsec.)
Figure 7. Numerical R vs. time of Al foam sample
Figure 8. Sequential images of an Al foam deformation during a SHPB test
Analysis of the Results
An increase in flow stress with strain rate has been noted in polymeric closed cell foams [19,20] and the increase in flow
stress, especially at high strain rates, is believed to be partly due to the inability of gas to escape from inside the cells and
partly due to strain rate sensitivity of the polymers. In the case of metal foams, and in open cell polymeric foams, any strain
rate dependence of the flow stress is principally due to the strain rate sensitivity of the matrix materials. The present foam does
not exhibit strain rate sensitivity, or at most only a very weak dependence which is masked by the scatter in experimental data.
It has been demonstrated that large deformations in foam samples at strain rates higher than 100 s-1 can be achieved In the
SHPB by means of recording several wave reflections as long as the duration of the input pulse and the lengths of the bars
permit separation of the waves. Zhao and Gary [21] have used a similar approach, but using a separation of wave’s technique,
to extend the usual strain range available in the SHPB, typically <20%, to the much higher strains necessary for investigation
of polymeric foams. The applied method can be used to construct the materials properties of the foams which will be an input
for modeling of foam parts in various kinds of application including blast mitigation and as filler in sandwich plates and
columnar structures. Similar method of reloading testing can further be used for the testing of light materials. When the
experimental data is coupled with numerical data, the variation of stress-equilibrium can be easily achieved. This is particularly
important for the validity of tests involving low density material testing in SHPB. Although it is not shown here, well-proven
numerical models can give variations of the stresses at various locations on/in the sample.
Conclusions
In this study, an Al-closed-cell foam was tested in SHPB by allowing the multiple passages of the waves. This multiple loading
allows testing the foam sample up to large strains without significantly reducing the strain rate between each loading cycle.
The tests were compared with quasi-static tests and it was found that the foam sample showed almost no strain rate
dependency of flow stresses. The variation of the stress equilibrium was calculated numerically. It was found that the stress
equilibrium variation decreases through the subsequent loadings, proving the increased mechanical impedance of the
deformed metal foam.
Acknowledgments
The authors would like to thank the Scientific and Technical Council of Turkey (TUBITAK) for the grant #MISAG-227 and NSFMISAG-7.
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