Materials Science & Engineering A 607 (2014) 498–504
Contents lists available at ScienceDirect
Materials Science & Engineering A
journal homepage: www.elsevier.com/locate/msea
Compressive properties of Advanced Pore Morphology (APM)
foam elements
M.A. Sulong a,c,n, M. Vesenjak b, I.V. Belova a, G.E. Murch a, T. Fiedler a
a
Centre for Mass and Thermal Transport in Engineering Materials, School of Engineering, The University of Newcastle, Callaghan 2308, Australia
University of Maribor, Faculty of Mechanical Engineering, Smetanova 17, SI-2000 Maribor, Slovenia
c
Department of Solid Mechanics and Design, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia – UTM, 81310 UTM Skudai, Johor, Malaysia
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 30 October 2013
Received in revised form
7 April 2014
Accepted 9 April 2014
Available online 16 April 2014
Advanced Pore Morphology (APM) foam elements have a spherical outer skin and a porous inner
structure. APM foam is a product of the improved powder metallurgy FOAMINALs process. The present
work investigates the mechanical properties of single APM foam elements under quasi-static and
dynamic compressive loading. By means of μCT techniques, an accurate finite element model is
generated. The compressive force–displacement response of APM foam elements is numerically
evaluated for different diameters and strain rates. The results of the numerical analysis are compared
with experimental data. Good agreement is found. Quasi-static and dynamic loading are both
investigated by making use of numerical analysis and verified by comparison with experimental results.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Finite element method
Mechanical characterisation
Tomography
Aluminium alloys
Plasticity
1. Introduction
Cellular materials are known for their very high porosity, high
specific stiffness, acoustic damping and the ability to absorb a
relatively high amount of energy at a low stress level [1,2]. Their
versatile heat insulation property makes them useful for heat
exchangers or thermal insulation [3]. A major limitation of conventional cellular materials is their stochastic geometry which can
result in unreliable mechanical properties. Another important
challenge is the reduction of their high manufacturing cost.
Control of the pore structure will allow the varying of the
morphology and topology (pore size distribution, pore shape and
cell wall geometry), which can be expected to allow a better
definition of mechanical properties [4,5]. Recently, a new concept
of metal foam components production was patented by Stö
bener
et al. [6]. Carrying the name Advanced Pore Morphology (APM, see
Fig. 1) foam, this innovative cellular material has been developed
on an improvement of the powder metallurgical FOAMINALs
process that was introduced earlier by Baumeister [7].
Several investigations have been conducted on either single or
composite APM foam elements with both partial and syntactic
morphology. The experiments have been carried out by Lehmhus
et al. [8] to investigate the influence of APM foam density for both
n
Corresponding author at: Centre for Mass and Thermal Transport in Engineering Materials, School of Engineering, The University of Newcastle, Callaghan 2308,
Australia.
E-mail address: mohdayub.sulong@uon.edu.au (M.A. Sulong).
http://dx.doi.org/10.1016/j.msea.2014.04.037
0921-5093/& 2014 Elsevier B.V. All rights reserved.
quasi-static and dynamic compressive loading as well as the effect
of varying the bonding agent (an epoxy-based adhesive and
polyamide). They reported that the dynamic initial strength of
epoxy-bonded APM foam is slightly higher than the quasi-static
initial strength. A notable increase in the dynamic initial strength
was observed for polyamide-bonded APM foam. Vesenjak et al. [9]
investigated the behaviour of a single APM foam element and a
composite APM foam with partial and syntactic morphology under
compressive loading. Two APM foam element sizes with diameters
∅¼ 5 mm and ∅¼10 mm were tested experimentally. The results
indicated that the larger APM spheres exhibit a higher energyabsorption capability due to a lower densification strain. They also
investigated cylinder-shaped (d¼h¼30 mm) epoxy samples with
embedded APM foam elements. Experimental testing was conducted using free and confined radial boundaries. It was found that
syntactic APM composites have energy absorption capacity approximately four times higher than non-bonded or partially bonded APM
foam elements. Hohe et al. [10] conducted experimental and
numerical tests on graded APM foams for multi-functional aerospace applications. The main focus of their investigation was
perforation resistance against bird strike events. In a case study, a
sandwich plate with graded APM foam core was compared with a
sandwich plate with a conventional foam core. The results indicated
that the use of a graded APM foam core increases the perforation
resistance performance of the sandwich plate. This was achieved by
dissipating the plastic energy over a larger volume. Vesenjak et al.
[11] used an infrared thermal imaging camera to enhance the usual
data acquisition during compressive experimental testing of APM
M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
Fig. 1. Light photograph of APM foam elements.
foam elements. Infrared thermal imaging indicated that plastic
yielding occurred predominantly in the outer region of the APM
foam element and then propagated inwards in a shear band into the
sphere. The present paper combines, for the first time, microcomputed tomography imaging and finite element analysis of
APM foam elements. In contrast with earlier investigations, a highly
accurate representation of the complex inner foam geometry is
obtained and incorporated in the numerical analysis. As a result,
detailed information on the internal deformation mechanisms is
obtained. For verification purposes, numerical results are compared
with experimental data published in the literature.
2. Manufacturing process of APM foam elements
The FOAMINALs process is capable of successfully manufacturing near net-shaped parts and three-dimensional (3D) sandwich
panels with a foamed core layer. In [10], an AlSi7 aluminium alloy
was prepared in powder metallurgy precursor form with TiH2
added as foaming agent. Foaming was activated by the heating of
the precursor. This process usually takes place within a mould
cavity. During the foaming process, furnace temperatures up to
800 1C are used depending on matrix alloys and the presence of
stabilising ceramic particles. As a result, the foaming mould has to
bear high thermal loads in each foaming cycle [8]. A new process
route proposed by Stö
bener in APM foaming eliminates the need for
expensive moulds. The manufacturing procedure consists of powder compaction (by the CONFORMs process) and rolling of the
AlSi7 alloy with TiH2 foaming agent to obtain expandable precursor
material. The wire-shaped precursor material (diameter ∅¼3 mm)
is then cut into small granulates (length l¼2 mm) [12], which are
expanded into sphere-like foam elements (see Fig. 1) due to the
thermal decomposition of the TiH2 foaming agent in a continuous
belt furnace. This is driven by internal gas pressure, surface tension
and the formation of an oxide skin. So far, three sizes of APM foam
elements have been manufactured with diameters of 5, 10, and
15 mm and foam element densities varying from 0.5 to 1.0 kg m 3
[8,13]. Structures assembled by APM foam elements exhibit two
types of porosity: (i) the inner porosity in single APM foam
elements and (ii) the outer porosity between many APM foam
elements which depends on the size of the APM elements and their
arrangement [14]. Detailed information on the automated production of APM foam elements can be found in the literature [12].
3. Model generation
Micro-computed tomography (μCT) imaging is a non-destructive
procedure that allows precise measurements on aluminium foams
499
[15–17] with the advantage of repetitive 3D assessment and computation of micro-structural and micro-mechanical properties. The
most outstanding feature of μCT is the ability to image the sample's
interior with high spatial and contrast resolution. A thorough review
of μCT can be found in the monograph [18]. In previous analyses,
sintered metallic hollow sphere structure (MHSS), sintered metallic
fibre structure (SMFS), lotus-type porous material, Alporass and MPores have been characterised using μCT technology [2,3,15–
17,19,20]. The companion study [21] first addressed μCT of APM
foam elements by analysing their geometrical micro- and mesostructures. By using the voxel data from the μCT images, geometrical
properties such as the number of pores and their size distribution in
the cellular materials were characterised. This novel approach can be
applied to any closed-cell morphology and will be expanded to
open-cell cellular materials. Results show that the deviation of the
porosity and the microstructure characteristics between specimens
of the same size was moderate. In the present study, μCT scanning of
small (∅¼ 5 mm) and large (∅¼ 10 mm) spherical foam elements
was performed. An Xradia MicroXCT-400 machine with a Hamamatsu L8121-03 X-ray source was used. Due to the size differences
of the samples, the voxel resolutions were limited to 3.61 μm
(∅¼5 mm sphere) and 6.28 μm (∅¼ 10 mm sphere), respectively.
A total of 1800 absorption radiographs (exposure time 16 s) was
captured with a 0.21 rotation for each projection. The selected
acceleration voltage was 140 kV with a current of 70 μA. The μCT
data was first segmented by gray-value thresholding. In this step, it
must be ensured that the volume of the voxels representing the
metallic phase matches the volume of metal in the APM foam
element. To this end, the mass of the scanned APM foam elements
was measured using a precision scale. Next, the metal volume was
calculated using the AlSi7 material density ρ ¼2680 kg m 3 [22].
The segmentation threshold was adjusted iteratively until the
segmented voxel volume coincided with this calculated volume.
The segmented voxels were then rendered to a 3D structure that was
subsequently converted into a stereolithography (STL) surface geometry mesh. Stereolithography surface geometry is widely used in
3D model rapid prototyping. Unlike the higher-level representation
such as non-uniform rational B-spline (NURBS) that contains a
smooth polynomial function or spline, the created STL mesh file is
not very rich in information [23]. STL meshes only describe the
surface geometry of a three-dimensional object using a triangulated
surface. In order to avoid the loss of geometrical detail during
meshing, at least 750,000 triangles were used in the STL surface
mesh. A volume mesh was then created within the bounds of the
surface mesh by using the commercial automatic mesh generation
software Sharc Harpoon. The small margin of error of the created
model mesh volume is found for both APM foam element sizes. The
small volumetric error percentages for ∅¼ 5 mm and ∅¼10 mm are
7% and 2%, respectively. A Hex-dominant mesh that contains three
types of elements, namely, linear hexahedral-, linear pentahedraland tethedral-elements have shown superior performance in the
numerical analysis of complex geometries [16]. Accordingly, mixed
meshes were used in the present analyses.
3.1. Finite element simulation
The accurate representation of the complex internal foam
geometry including the micro and macro-porosity requires a high
geometrical resolution. As a result, a single APM foam element is
too large to be meshed (and computed) on the available large
memory computers (64 GB RAM). In this work, we made use of the
symmetry of the roughly spherical-shaped APM foam elements to
reduce the size of the calculation model. To this end, each
spherical APM element was subdivided into eight segments (see
Fig. 2). For each diameter (i.e., 5 mm and 10 mm) four segments
were converted into numerical calculation models.
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M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
Fig. 2. Sub-division of APM foam elements.
Fig. 3. Boundary conditions of the finite element analyses.
The boundary conditions of the finite element analysis are
shown in Fig. 3. A rigid support was defined as part of a contact
boundary condition with the deformable elements of the APM
volume meshes. Quasi-static and dynamic compressive loading
was introduced using a time-dependent nodal displacement
boundary condition at the upper symmetry plane of the foam
element. For the dynamic loading case the rate of the displacement boundary condition was adjusted to achieve a macroscopic
strain rate of 100 s 1. Within the remaining two symmetry planes,
the normal displacement was set to zero in order to simulate the
remaining APM foam element. Such boundary conditions were
introduced with the assumption that the deformation propagates
in an identical way at the top and the bottom of the foam
elements. Nevertheless, in the actual experiment the deformation
progress can only be observed and confirmed at the top of the
foam elements since the moving force platen is located at the top.
The maximum deformation of the model was set to 10% (quasistatic) and 70% (dynamic) of the original APM sphere's height. The
smaller deformation of the quasi-static model is caused by
significantly longer calculation times. Material properties of AlSi7
used in this numerical model were Young's modulus
E¼ 72,400 MPa, Poisson's ratio ν ¼0.3, yield stress sy ¼120 MPa,
density ρ ¼ 2680 kg m 3, and tangent modulus T ¼800 MPa [22].
Within the dynamic simulations the base material inertia and its
strain rate effect have been accounted for [20]. The dynamic
response of cellular materials is mainly influenced by (i) the
micro-inertial effect and (ii) the material strain-rate sensitivity
[24–26]. The micro-inertia of the individual cell walls can affect
the deformation of metal foams as discussed in [27,28]. According
to Tan et al. [29] the inertia effects associated with the dynamic
localisation of crushing are responsible for the enhancement of the
dynamic strength properties in high velocity regimes. The authors
in [28] state that inertia effects contribute significantly to the rate
effect, even when the strain rate sensitivity of material properties
is ignored. The material rate-dependence is usually taken into
account within the constitutive material model [30]. In addition,
the strain-rate sensitivity of the cellular structure might be
affected by the presence of a gaseous pore filler (e.g., air, H2)
[31,32]. However, the strain rate sensitivity effect is usually more
apparent for structures with higher initial pore pressures or higher
relative densities [33,34]. It should be noted that air compression
within the cells was not considered in the scope of this work
which might influence the response of the APM element subjected
to dynamic loading (high strain rates). The base material strain
rate sensitivity was introduced using the Cowper–Symonds constitutive model with the parameters C ¼6500 s 1 and p ¼4
[35–37]. All nodes of the aluminium foam belong to a single
surface contact accounting also for friction with a coefficient of
friction of 0.5 [38]. A user-defined subroutine was used to extract
force–displacement data. To this end, the sum of nodal reaction
forces and the average displacement of these nodes selected on
the upper plane were recorded. The computational models were
analysed using the commercial finite element codes MSC.Marc for
the quasi-static loading case and LS-DYNA for the dynamic loading
case [37].
It should be mentioned here that the simulation of a complete
APM foam element will serve better results. The simulation of a
full sphere with an adequate geometrical resolution that captures
the internal pore structure requires extremely long computation
times (44 months) and large random access memory (4 128 GB).
This obstacle is overcome by considering only a segment (oneeighth of an APM foam element) for the numerical simulation.
However, this simplification introduces a number of shortcomings
that should be considered during the result interpretation. First,
the size and shape of pores that intersect a symmetry plane are
going to be altered. Second, the boundary conditions may affect
local bending and buckling of the solid phase. However, this effect
seems to be limited since the analysis of stress and strain
distributions in the vicinity of and away from the boundary
condition showed no significant deviation. Third, the boundary
conditions affect the stress wave propagation through the foam
structure and influence the dynamic response. However, considering the high porosity of the structure and moderate loading
velocities the influence of the symmetrical boundary is not
significant. This has also been confirmed by validation of the
computational results with experimental tests.
Prior to the mechanical characterisation of APM foam elements,
the mesh independence of the solution was addressed. The results
of the mesh refinement study are illustrated in Fig. 4. The
boundary conditions described above were used and the initial
gradient dF/dy of the force–displacement curve was calculated.
Fig. 4 shows that 640,000 nodes (∅¼5 mm) and 550,000 nodes
(∅¼10 mm) yielded satisfactory numerical convergence. The corresponding average element sizes were 0.014 mm for ∅¼ 5 mm
and 0.035 mm for ∅¼10 mm APM foam elements. In addition, for
the dynamic simulations, the time step sensitivity analysis had to
M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
be performed in order to satisfy the Courant–Friedrichs–Lewy
condition for the explicit integration scheme. Based on the
smallest finite element edge length it was confirmed that the
time step of 0.39 μs provides stable convergence [36].
4. Results and discussion
Fig. 5 shows the quasi-static force–displacement response of
APM foam elements with ∅¼5 mm and ∅ ¼10 mm diameters.
Numerical simulations of APM foam elements are denoted by thin
Fig. 4. Convergence analysis on APM foam elements.
Fig. 5. Quasi-static response for (a) ∅¼ 5 mm APM foam and (b) ∅¼ 10 mm APM
foam elements.
501
continuous lines and their average value is represented by a thick
continuous line. The experimental data taken from [9] and the
corresponding average are denoted by the thin and thick dotted
lines, respectively. It can be observed that absolute values as well
as the scattering of numerical and experimental results are
consistent. In particular, good agreement is found in their averaged
values. The comparison of large (∅¼10 mm) and small (∅¼5 mm)
foam elements clearly indicates higher compressive force values
for the large APM element. This result is expected since (unlike
stress–strain data) force–displacement data are not normalised by
the sample size. Accordingly, the larger (∅¼10 mm) foam elements provide more resistance to compressive loading. To enable a
direct comparison between the two foam element sizes, additional
numerical simulations are conducted. To this end, two of the 3D
segments from the ∅¼ 5 mm APM elements are scaled in size by a
factor of two (i.e., their diameter is increased to 10 mm). This
scaling process does not change the geometry of the microstructure of the ‘small’ APM spheres. It can be observed that the force–
displacement curves from the “scaled ∅¼5 mm” APM foam
elements are different from the ∅¼10 mm APM spheres. The
“scaled ∅¼5 mm” curves exhibit a reduction in structural stiffness
between displacements of 0.2–1.0 mm. The most likely explanation for this deviation is differences in the internal microstructure
and outer particle shape between “scaled ∅ ¼5 mm” and 10 mm
APM foam elements.
The structural stiffness, k of ∅¼ 5 mm and ∅¼10 mm APM
foam elements is shown in Table 1. It can be seen from Fig. 5 that
APM foam elements with different diameters exhibit different
stiffnesses. The structural stiffness is the change in compression
force divided by the difference in displacement, e.g., the slope of
the dot-dashed line in Fig. 5b. For the ∅¼10 mm APM foam
elements, the displacement range between 0.2 mm and 1.0 mm
was considered. Low displacements are disregarded in order to
eliminate initial effects due to the establishment of contact
between the foam element and contact plane. In the case of the
∅¼5 mm APM foam elements, the selected range is 0.15–0.5 mm.
The average structural stiffness of ∅¼10 mm APM foam elements
is 3.3 times higher in comparison to ∅¼5 mm APM foam elements. In addition, the relative variability value was calculated by
dividing the standard deviation of the structural stiffness by the
corresponding average value. Comparing the relative variability of
both diameters, it can be seen that the structural stiffness of the
larger foam elements is more consistent. The variability of the
stiffnesses is probably linked to the relative size of the pores with
respect to the diameter of the foam elements. A study has been
conducted in [21] to investigate the number of pores in one
∅¼5 mm and one ∅ ¼10 mm APM element with respect to their
pore radius. It has been found that the number of pores per APM
element for ∅¼5 mm and ∅¼10 mm is 4363 and 13,797, respectively. Furthermore, the study conducted in [21] also counted the
number of large pores in ∅ ¼5 mm and ∅¼10 mm APM foam
elements. These largest pores in the APM (with radius of about
20% or larger of the foam elements' diameter) are likely to strongly
affect the mechanical properties of APM foam elements. Referring
to the data reported in [21], there are more than 10 pores larger
than 2 mm in radius for the ∅¼10 mm APM foam element.
Table 1
Structural stiffness of ∅¼ 5 mm and ∅¼ 10 mm APM foam elements determined by the slope of their respective graphs.
APM sphere size (mm)
∅ ¼5
∅ ¼10
Effective stiffness, k (N/mm)
Simulations
Segment #1
Segment #2
Segment #3
Segment #4
Average
Relative variability
107.4
370.0
166.6
412.2
76.6
328.7
77.6
280.9
107.0
348.0
0.39
0.16
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M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
Furthermore, there are more than 120 pores larger than 1 mm in
radius distributed in the ∅¼10 mm APM sphere. It should also be
noted that the largest pores (radius Z20% of the foam elements'
diameters) shown in Figs. 1, 2, 6, 7 and 9 are non-homogenously
distributed compared to the more evenly distributed pores of size
1 mm in radius. Investigation done in [21] also found more than 10
pores (with a radius of about 20% or larger of the foam elements'
diameter) in a ∅¼5 mm APM sphere, but the number of pores
with radius of about 10% of the foam elements' diameters is much
lower (fewer than 20 pores). The higher number of more evenly
distributed pores (pores with radius Z10% of the foam elements'
diameters) in ∅¼ 10 mm APM sphere results in more consistent
results. On the other hand, a significant reduction of the number of
pores with radius attaining more than 10% of the foam elements'
diameters in a ∅¼ 5 mm APM foam element has some effect in
representing an adequate RVE and this resulted in inhomogeneity
of the stiffnesses.
The scatter of the structural stiffness for ∅¼5 mm APM sphere
was also possibly caused by the irregular outer shape of the
simulated segments. Comparing all the segments from ∅ ¼5 mm
to ∅ ¼10 mm APM foam elements, the outer shape of ∅¼10 mm
APM segments is closer to a perfect sphere. In contrast, the smaller
APM foam elements exhibit a more irregular outer shape potentially causing scatter of their mechanical properties.
An additional numerical simulation was conducted to investigate the effect of the skin on the APM foam sample. To this end,
the skin has been virtually removed from segment #2 of the
∅¼10 mm APM foam sphere. This reduced the sphere volume by
13%. The structural stiffness of both APM foam elements with and
without skin was then calculated. The results of 412.2 N/mm for
the original sphere and 290 N/mm for the sphere without the skin
reveal that a significant reduction of structural stiffness has
occurred. The structural stiffness of the sphere without the skin
is reduced by a factor of 1.4 compared to the original APM sphere
elements. The skin thickness is more for diameter 5 mm APM foam
elements compared to the diameter 10 mm APM foam element.
Fig. 6 shows the distribution of equivalent plastic strain for
quasi-static loading for segment #1 of ∅ ¼5 mm and ∅ ¼10 mm
diameter APM foam spheres. The maximum displacement applied
is limited to 10% of the sample height. This displacement was
chosen in order to achieve an adequate calculation time. Thus, for
the APM foam element with ∅¼5 mm and ∅¼10 mm, the applied
displacement is 0.25 mm and 0.5 mm, respectively. It can be
observed that the upper part of the foam element is flattened by
the simulated compressive plate. This is more likely due to
minimum cross section. A similar observation has been reported
by Stö
bener et al. [39] who conducted an experimental compression test of ∅¼10 mm APM foam elements. Fig. 6 further shows
that plastic deformation is not uniformly distributed over the
complete foam element but occurs predominantly in proximity of
the contact area. Vesenjak et al. [11] studied the propagation of the
plastic yielding within the APM foam element using infrared
measurements. They too observed that the yielding originates
from the contact area between the APM foam and the rigid
support and spreads in a shear band with an angle of 451 towards
the lower part of the foam element. Fig. 6 clearly supports this
observation and demonstrates that plasticisation is not limited to
the immediate surface area. A line with a distance of 2.5 mm from
the initial contact point is drawn in Fig. 6a and b. It can be seen
that independent of the foam element size a similar depth is
affected by high levels of plastic deformation at the same macroscopic deformation of u ¼ 0.25 mm. However, in the case of the
larger foam elements the affected volume is larger.
4.1. Internal deformation analysis
Fig. 6. Distribution of total equivalent plastic strain for macroscopic deformation of
u¼ 0.25 mm: (a) APM foam element with ∅¼5 mm, and (b) APM foam element
with ∅¼10 mm (contours represent the total equivalent plastic strain).
The internal deformation characteristics of APM foam elements
under quasi-static loading are elaborated here. Localised concentrations of equivalent plastic strain are shown in Fig. 7. The
distribution of plastic strain validates the shear band premise
made earlier in this section. During compression, a shear cone
Fig. 7. Common features of internal distortion under compressive load.
M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
with an angle of 451 (generated by revolving the 451 line about the
vertical axis in Fig. 6) with respect to the loading axis was formed.
Whenever features like thin walls and thin struts are located
within the shear cone, they will be among the first to experience
high levels of plastic deformation. In addition, high levels of
plasticity occur near the contact area of the external load. In
quasi-static analysis, this phenomenon is caused by the high
contact stresses (contact force divided by contact area) due to
the small initial contact area of the sphere. However, in dynamic
analysis the flattening of the top part of the model is (regardless to
the loading velocity) a result of the distribution of cross sectional
areas in sphere-like bodies. Such behaviour is different from the
uni-axial compression observed in hexahedral shape cellular
material (e.g., Alporass), wherein the incremental deformation
maps show collapse bands originating from neighbouring previous
bands [40].
Next, the dynamic behaviour of APM foam elements is considered. The time step increment in these direct finite element
simulations is based on the size of the shortest finite element
edge, i.e., 0.0093 mm for ∅¼ 5 mm foam and 0.0109 mm for
Fig. 8. Numerical simulations and experimental results for ∅¼ 10 mm APM foam
element.
503
∅¼10 mm foam. These small lengths put extreme demands on
computational time. As a result, dynamic analyses were limited to
∅¼10 mm foam elements. Fig. 8 shows the dynamic force–
displacement response of ∅¼10 mm APM foam elements. Numerical simulations of APM foam elements were conducted at a
macroscopic strain rate of 100 s 1 and are denoted by thin
continuous lines. Their average value is represented by a thick
continuous line. In addition, experimental data taken from [11] are
shown. The thick dotted line shows the results of experimental
testing with a macroscopic strain rate of 20 s 1. For comparison,
results of quasi-static testing are plotted as a dashed line. The
compressive response of APM foam elements shows the typical
behaviour of porous materials. Following the initial elastic
response one can observe the onset of an elastic–plastic transition
zone that is then manifested as a plateau followed by the final
densification [41]. The averaged numerical results are in good
agreement with the experimental measurements. The numerical
results correspond to a compressive strain rate 100 s 1. In contrast, experimental results were obtained at lower strain rates (i.e.,
20 s 1) and quasi-static conditions. Numerical tests had to be
performed at the higher strain rate since a transition to 20 s 1
would have caused unreasonably long calculation times. As a
result, the numerical results exhibit an increased plateau stress
and slightly decreased densification strain. Furthermore, it can be
observed that the stiffness of the structure increases at high
loading rates.
Fig. 9 shows the distribution of total equivalent plastic strain in
the ∅¼ 10 mm APM foam element at different displacement
increments. As already observed in experiments and quasi-static
numerical results, the local deformation originates at the contact
between the APM foam element and the loading/support plate due
to the lowest load bearing cross section of the spherical-shaped
object. Again we can note that the propagation of the plastic strain
spreads in a shear band towards the centre of the foam element in
agreement with previous observations [11]. The numerical investigations conducted within this paper only focus on a single APM
sphere rather than on a more complex structure made up of
Fig. 9. Distribution of total equivalent plastic strain for the ∅¼ 10 mm APM foam element with contours (contours represent total equivalent plastic strain).
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M.A. Sulong et al. / Materials Science & Engineering A 607 (2014) 498–504
several APM spheres. Assuming that one APM sphere can be
considered as a representative volumetric element (RVE) of a
structure formed by these elements, the findings presented in this
work can to some extent be used to predict the mechanical
properties of this material.
5. Conclusions
This paper has addressed the compressive properties of spherical APM foam elements. Two different sphere sizes, i.e., ∅ ¼5 mm
and ∅¼10 mm were investigated. Numerical finite element analyses were conducted for quasi-static and dynamic loading. To
account for the complex interior foam geometry, calculation
models were directly derived from micro-computed tomography
data. Results were compared with experimental measurements
conducted on similar samples. Good agreement was found. Single
APM foam elements demonstrated typical cellular material behaviour, i.e., an extended stress plateau followed by densification at
high macroscopic strains. Numerical analyses of the APM mesostructure showed that plastic deformation does not occur uniformly but is concentrated in a relatively small sub-volume. This
deformation is not limited to the surface of the spherical element
but continues within the interior foam structure. More specifically,
plasticity originates from the contact area of pressure stamp and
sphere and propagates in a 45o shear cone towards the interior of
the samples. As expected, larger APM foam elements were able to
support higher compressive loads, i.e., ∅¼10 mm APM foam
elements exhibit a 3.3 times higher structural stiffness. The
investigation on the skin effect shows that the skin significantly
contributes to the mechanical strength of APM foam elements. The
single APM foam elements with removed skin have a reduction in
structural stiffness by a factor of 1.4 from the original APM foam
counterpart. Internal microstructure deformation analysis reveals
that weak structures like thin walls and thin struts within the 451
shear cone are likely to experience high levels of plastic deformation. Finally, dynamic compression simulation indicates an
increase of stiffness and plateau stress with increasing macroscopic strain rate.
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