Static and dynamic crushing of circular aluminium extrusions with aluminium foam filler.pdf

* Corresponding author. Tel.: 0047-73-594782; fax: 0047-73-594701.
E-mail addresses: arve.hanssen@bygg.ntnu.no (A.G. Hanssen), magnus.langseth@bygg.ntnu.no (M. Langseth)
International Journal of Impact Engineering 24 (2000) 475}507
Static and dynamic crushing of circular aluminium extrusions
with aluminium foam "ller
A.G. Hanssen*, M. Langseth, O.S. Hopperstad
Structural Impact Laboratory (SIMLab), Department of Structural Engineering, the Norwegian University of Science
and Technology, N-7491 Trondheim, Norway
Received 15 March 1999; received in revised form 28 October 1999
Abstract
An experimental programme consisting of 96 tests was carried out to study the axial deformation
behaviour of triggered, circular AA6060 aluminium extrusions "lled with aluminium foam under both
quasi-static and dynamic loading conditions. The outer diameter and length of the columns were kept
constant at 80 mm and 230 mm, respectively. The main parameters in addition to the loading condition were
the foam density, the extrusion wall strength and the extrusion wall thickness. Based on the experiments,
design formulas for prediction of average force, maximum force and e!ective crushing distance were
suggested. ( 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction
During the last two decades, the transportation industry has focused their attention on vehicle
behaviour during crash events. New legislation coupled with important marketing advantages has
emphasised the need for structural systems with documented crashworthiness integrity. Further-
more, requirements on CO
2
emissions and the desire for cars with economical handling put
additional restrictions on the vehicle weight. Typical actions needed to ful"l the total set of
requirements are redesigned structural elements, often incorporating new materials.
Axial crushing of thin walled, circular tubes made of mild steel was recognised by Alexander [1]
as an excellent mechanism for energy absorption. Furthermore, the simple geometry of circular
tubes coupled with an assumed perfect plastic material behaviour enabled Alexander [1] to deduce
a simple expression for the average crushing force. Several improvements to this solution have been
0734-743X/00/$- see front matter ( 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 1 7 0 - 0

Nomenclature
d displacement
d
max
displacement at maximum value of ¹
E
l component length
b outer component cross-section diameter
h component wall thickness
b
m
b
m
"b!h
b
i
b
i
"b!2h
F crush force
F
avg
, F
max
average and maximum crush forces in interval [0, d]
E absorbed energy
A
E
crush force e$ciency
¹
E
total e$ciency
D
C
deformation capacity, relative deformation
S
E
e!ective crush distance, stroke e$ciency, D
C
(d
max
)
p stress, engineering
p
0.2
extrusion wall stress at 0.2% plastic strain
p
U
extrusion wall ultimate stress
p
0
extrusion wall characteristic stress, 0.5(p
0.2
#p
U
)
p
f
foam plateau stress
e strain, engineering
m, n material constants
C
pow
power law coe$cient
o
0
extrusion material density
o
f
, o
f0
foam density and foam base material density
o
fC
critical foam density giving transition from diamond to concertina mode
C
avg
, C
max
interaction constants of average and maximum force
C
SE
stroke e$ciency constant
a, b dimensionless constants
v
0
impact velocity
M impacting mass
(EP
kin
#EP
pot
)/E
calc
control parameter, energy absorption
d
real
/d
calc
control parameter, permanent displacement
suggested over the years, e.g. [2}5]. The functionality of tubes with other cross sections has also
been veri"ed and proper solutions for the average crush force developed [6]. As a means of
improving the energy absorption capabilities of thin walled tubes, columns with foam "ller
have been investigated [7}12]. Typical for the earlier work is the focus on steel tubes "lled
with polyurethane foam "ller. Some theoretical considerations developed in parallel with these
experiments can be found in [9}11]. Recently, the focus has been turned towards utilisation of
476 A.G. Hanssen et al. / International Journal of Impact Engineering 24 (2000) 475}507

aluminium foam "ller. New and cost e!ective manufacturing methods in addition to high strength
to weight ratio makes the use of these foams attractive for commercial use. Seitzberger et al. [13]
carried out a limited experimental study of mild steel tubes with aluminium foam "ller of both
square and circular cross sections. A common feature of the recent work on foam "lled columns is
the limitation to square cross sections. Refs. [14}16] presented extensive experimental data on
aluminium extrusions with aluminium foam "ller, whereas Santosa et al. [17,18] and Hanssen et al.
[19] utilised some of these data for validation of numerical codes. Based on design formulas
developed from their experimental programme, Hanssen et al. [20] was able to show that
signi"cant weight savings are possible by utilisation of aluminium foam "ller. A consequence of the
general solution is that the outer cross-section of the foam "lled columns has to be reduced
compared with traditionally designed non-"lled columns if increase of mass speci"c energy
absorption is to be achieved.
The current experimental programme is a natural extension of the experimental investigation
previously carried out by Hanssen et al. [16] on square cross-sections. All components tested
herein had a circular cross-section with an outer diameter of 80 mm and were made from
extrusions of the aluminium alloy AA6060. The objective was to investigate the e!ect of extrusion
wall thickness, extrusion material strength and foam "ller density on the crushing characteristic of
these columns subjected to both quasi-static and dynamic loading conditions. All components
tested had a trigger at the top to initiate the deformation pattern during axial loading.
2. Terminology of axial crushing
This section follows the approach in Hanssen et al. [16]. Fig. 1 depicts the characteristic
behaviour of energy absorbing components and summarises the corresponding terminology
applied herein.
The relative deformation, being the actual deformation d divided by the original length l of the
crushed component, is termed the deformation capacity D
C
. The ratio between the average force
F
avg
and maximum force F
max
, both calculated in the interval M0, dN, is referred to as the crush force
e$ciency A
E
of the absorber and can never reach values above 100%. A given energy absorber may
be considered to show optimum properties when the total energy absorbed E divided by the
maximum force F
max
obtains a maximum value. This property is represented by the total e$ciency
¹
E
, see Fig. 1. The corresponding maximum value of the total e$ciency ¹
E
occurs at a deformation
of d
max
. Equivalently, the relative deformation D
C
at which the maximum value of ¹
E
occurs is
referred to as the stroke e$ciency or e!ective crushing distance S
E
of the absorber. Hence, utilising
the absorber for deformation capacities larger than that represented by the stroke e$ciency S
E
has
no sense since the corresponding force levels reaches unrealistically large values. As shown in Fig. 1,
the total e$ciency ¹
E
may be computed as the product of the crush force e$ciency A
E
and the
deformation capacity D
C
.
3. Experimental test programme
The target of this experimental programme was to investigate the force deformation behaviour of
foam "lled extrusions. To simplify the analysis of the data, it was found convenient to select some
A.G. Hanssen et al. / International Journal of Impact Engineering 24 (2000) 475}507 477

Fig. 1. Terminology applied for axial crushing of column [14,16].
response parameters directly relating to the characteristics of the force deformation behaviour.
Hence, the main response parameter of interest in the current programme was the average crush
force F
avg
. Second on the list were the maximum force F
max
and the e!ective crushing length S
E
.
The main model parameter whose in#uence was examined was the density of the foam "ller.
However, the experimental programme must be based upon how the foam "ller interacts with the
other model parameters and thus in#uences the above selected response parameters. As seen from
Table 1, three di!erent foam densities (o
1
, o
2
and o
3
) were investigated in addition to the non-"lled
columns (o
f
"0). The other model parameters and the corresponding number of test levels were
the loading condition (2 levels: S and D), extrusion wall thickness (3 levels: h
1
, h
2
and h
3
) and
extrusion wall strength (2 levels: p
1
and p
2
). All possible levels of the di!erent model parameters
were combined, resulting in a total of 4]2]3]2"48 characteristic tests. This programme was
carried out two times (a and b), giving a total of 96 tests.
Concern was given only to extrusions with a circular cross-section, having a constant outer
diameter equal to 80 mm and an e!ective length of 230 mm, see Fig. 4 in Section 5. The three
di!erent levels of the wall thickness showed average values of h
1
"1.43, h
2
"1.98 and
h
3
"2.36 mm, whereas the average foam densities were o
1
"0.13, o
2
"0.25 and o
3
"0.35 g/cm3.

Table 1
Experimental test programme
Parameters Repetition
Cross-section Loading condition Wall strength Wall thickness Foam density
Circular (C) Static (S) p
1
h
1
o
f
"0 a
Dynamic (D) p
2
h
2
o
1
b
h
3
o
2
o
3
No. of levels: 1 2 2 3 4 2
Half the test programme was carried out by quasi-static loading S, whereas the remaining
specimens were subjected to dynamic loading D at impact velocities ranging from 12 to 23 m/s.
Tables 2 and 3 show values of the model and response parameters for all tests. Here the following
test numbering system has been applied, e.g. CD132b corresponds to C: circular extrusion, D:
dynamic loading condition (S is applied for static case), 1: material property p
1
, 3: wall thickness h
3
,
2: foam density o
3
and "nally b: repetition b.
4. Material properties
4.1. Aluminium foam
Hydro Aluminium a.s. manufactured the aluminium foam sheets from which the test specimens
in the current test programme were machined. Three batches of foam with average densities of
o
1
"0.13, o
2
"0.25 and o
3
"0.35 g/cm3 were delivered. Cylindrical specimens with both dia-
meter and length equal to 80 mm were prepared from each batch of foam in order to carry out
uniaxial material testing. Care was taken to machine all foam specimens so that their axial
direction corresponded to the in-plane and normal-to-casting direction (transverse) of the manu-
factured foam sheets. This direction also corresponded to the axial direction of the foam "ller used
in the main test programme, Table 1. Owing to various process parameters and the in#uence of
gravity, the produced foam sheets show anisotropic material behaviour. Hence, unnecessary scatter
in the test results is avoided by machining all foam specimens from the same direction. Fig. 2 shows
the results from the static compression tests. The power law [21]
p
f
"C
powC
o
f
o
f0
D
m
(1)
relating foam plateau stress p
f
to foam density o
f
has been "tted to the statically obtained
experimental data by use of the method of least squares. The corresponding values of the
coe$cients C
pow
and m are also given in this "gure. In Eq. (1), the foam base material density is

Table 2
All static tests
Test No. Extrusion Foam Response parameters
Width Thickness Yield stress Ultimate Charact. Density Plateau Max. force Mean force Stroke Total no.
b h p
0.2
stress stress o
f
stress at D
C
"50% at D
C
"50% e$ciency of lobes
(mm) (mm) (MPa) p
U
(MPa) p
0
(MPa) (kg/dm3) p
f
(MPa) F
max
(kN) F
avg
(kN) S
E
(%) and types!
cs110a 80.0 1.40 72 165 119 0 0 27.9 16.0 78.2 7}8 D
cs110b 80.0 1.48 72 165 119 0 0 31.2 17.7 76.2 8 D
cs111a 80.0 1.40 72 165 119 0.13 0.4 30.3 21.0 75.1 9 D
cs111b 80.0 1.41 72 165 119 0.17 0.9 32.5 24.3 74.0 9 D
cs112a 80.0 1.40 72 165 119 0.26 3.1 53.5 41.3 68.3 19 C
cs112b 80.0 1.41 72 165 119 0.25 2.9 56.6 38.1 73.9 19 C
cs113a 80.0 1.41 72 165 119 0.35 8.1 76.9 64.1 57.8 19 C
cs113b 80.0 1.40 72 165 119 0.35 7.9 73.3 60.0 66.3 19 C
cs120a 80.0 1.99 79 169 124 0 0 44.2 28.0 73.0 6 D
cs120b 80.0 1.97 79 169 124 0 0 40.4 27.0 72.9 6 D
cs121a 80.0 1.97 79 169 124 0.14 0.5 53.0 35.3 70.0 7 D
cs121b 80.0 1.97 79 169 124 0.15 0.6 54.0 35.6 70.4 7 D
cs122a 80.0 1.97 79 169 124 0.24 2.6 77.8 50.7 72.1 15 C
cs122b 80.0 1.97 79 169 124 0.23 2.1 72.5 54.0 66.7 15 C#D
cs123a 80.0 1.96 79 169 124 0.34 7.4 94.9 75.4 67.5 15 C
cs123b 80.0 1.98 79 169 124 0.37 9.5 98.2 80.4 59.7 17 C
cs130a 80.0 2.38 71 157 114 0 0 50.8 36.4 70.6 6 D
cs130b 80.0 2.46 71 157 114 0 0 64.0 39.9 76.1 5 D
cs131b 80.0 2.39 71 157 114 0.11 0.3 64.5 42.8 69.9 6 D
cs132a 80.0 2.36 71 157 114 0.25 2.8 82.7 59.2 70.8 13 C
cs132b 80.0 2.46 71 157 114 0.28 4.0 92.9 68.0 67.2 15 C#D
cs133a 80.0 2.37 71 157 114 0.35 7.7 113.5 85.3 64.2 13 C
cs133b 80.0 2.39 71 157 114 0.37 9.2 115.2 92.5 64.0 13 C
cs210a 80.0 1.48 110 176 143 0 0 33.5 20.3 78.6 8 D
cs210b 80.0 1.42 110 176 143 0 0 30.4 18.8 78.7 8}9 D
cs211a 80.0 1.42 110 176 143 0.14 0.6 38.7 24.2 74.7 9 D
cs211b 80.0 1.41 110 176 143 0.15 0.7 34.8 25.3 73.5 9 D
cs212a 80.0 1.41 110 176 143 0.26 3.4 57.9 43.7 69.1 21 C
cs212b 80.0 1.41 110 176 143 0.27 3.6 64.2 45.6 74.7 21 C
cs213a 80.0 1.41 110 176 143 0.33 6.6 76.1 64.0 66.6 19 C
480
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475}507

cs213b 80.0 1.48 110 176 143 0.32 6.0 80.3 63.7 71.1 19 C
cs220a 80.0 1.98 132 195 164 0 0 78.5 37.0 72.2 6 D
cs220b 80.0 1.96 132 195 164 0 0 63.6 35.3 76.6 7 D
cs221a 80.0 1.98 132 195 164 0.13 0.4 62.3 40.7 73.9 7 D
cs221b 80.0 1.98 132 195 164 0.14 0.6 57.2 40.6 73.9 7 D
cs222a 80.0 1.99 132 195 164 0.25 2.9 99.1 63.2 72.5 15 C#D
cs222b 80.0 1.98 132 195 164 0.22 2.0 94.4 61.1 73.1 13 C
cs223a 80.0 1.98 132 195 164 0.37 9.5 119.9 90.1 67.1 13 C
cs223b 80.0 1.98 132 195 164 0.37 9.1 112.9 85.7 63.6 13 C
cs230a 80.0 2.37 126 187 157 0 0 89.9 48.4 75.1 5 D
cs230b 80.0 2.38 126 187 157 0 0 78.9 45.9 75.6 6 D
cs231a 80.0 2.36 126 187 157 0.10 0.2 81.0 46.8 70.2 5 D
cs231b 80.0 2.45 126 187 157 0.16 0.8 96.5 57.9 73.0 6 D
cs232a 80.0 2.37 126 187 157 0.24 2.6 104.0 71.1 75.4 15 C
cs232b 80.0 2.45 126 187 157 0.24 2.7 113.4 73.3 75.2 15 C
cs233a 80.0 2.36 126 187 157 0.32 6.1 111.9 90.2 69.4 15 C
cs233b 80.0 2.36 126 187 157 0.34 7.4 130.4 94.6 63.0 15 C
!De"nition after [5]: C"concertina, D"diamond, Global bending: b.
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481

Table 3
All dynamic tests
Test no. Extrusion Foam Initial velocity, accuracy control and response parameters
Width Thickness Yield stress Ultimate Charact. Density Plateau Initial Mean force EP
kin
#EP
pol
E
calc
d
real
d
calc
Total no.
b h p
0.2
stress stress o
f
stress velocity at max. D
C
of lobes
(mm) (mm) (MPa) p
U
(MPa) p
0
(MPa) (kg/dm3) p
f
(MPa) v
0
(m/s) F
avg
(kN) (!) (!) and types!
cd110a 80.0 1.40 72 165 119 0 0 11.9 21.5 1.013 0.935 8 D
cd110b 80.0 1.42 72 165 119 0 0 11.9 22.2 1.016 0.968 8 D
cd111a 80.0 1.41 72 165 119 0.14 0.5 12.7 26.7 1.014 0.968 9 D
cd111b 80.0 1.41 72 165 119 0.12 0.3 12.8 26.4 1.013 0.970 9 D
cd112a 80.0 1.40 72 165 119 0.25 3.0 16.1 48.6 1.024 1.027 19 C
cd112b 80.0 1.49 72 165 119 0.27 3.7 16.2 50.4 1.014 0.968 17 C
cd113a 80.0 1.42 72 165 119 0.36 8.3 18.3 72.3 1.017 1.013 17 C b
cd113b 80.0 1.41 72 165 119 0.32 6.0 " 18.3 63.5 } } 19 C b
cd120a 80.0 1.98 79 169 124 0 0 13.6 33.7 1.017 1.009 6 D
cd120b 80.0 2.01 79 169 124 0 0 13.5 31.9 1.014 0.974 6 D
cd121a 80.0 1.99 79 169 124 0.14 0.5 15.1 40.5 1.018 1.035 6 D
cd121b 80.0 1.97 79 169 124 0.11 0.3 15.2 39.2 1.018 0.999 10 D#C
cd122a 80.0 2.01 79 169 124 0.24 2.6 17.8 62.2 1.022 1.077 15 C
cd122b 80.0 1.98 79 169 124 0.27 3.4 " 17.8 58.2 } } 15 C
cd123a 80.0 1.97 79 169 124 0.35 7.6 20.4 85.4 1.048 1.036 15 C b
cd123b 80.0 1.98 79 169 124 0.33 6.8 20.6 84.1 1.024 1.061 15 C
cd130a 80.0 2.38 71 157 114 0 0 15.6 43.3 1.016 1.012 6 D
cd130b 80.0 2.45 71 157 114 0 0 15.6 44.6 1.016 1.004 6 D
cd131a 80.0 2.38 71 157 114 0.12 0.3 16.2 49.6 1.019 1.030 6 D
cd131b 80.0 2.47 71 157 114 0.13 0.4 16.1 50.5 1.017 1.022 7 D
cd132a 80.0 2.38 71 157 114 0.23 2.4 19.8 73.9 1.025 1.060 13 C
cd132b 80.0 2.36 71 157 114 0.24 2.6 19.7 71.1 1.016 0.983 13 C#D
cd133a 80.0 2.37 71 157 114 0.35 8.1 " 21.6 96.7 } } 15 C b
cd133b 80.0 2.36 71 157 114 0.34 7.4 " 21.7 92.2 } } 13 C
cd210a 80.0 1.48 110 176 143 0 0 12.1 24.9 1.013 0.964 8 D
cd210b 80.0 1.39 110 176 143 0 0 12.2 22.4 1.014 0.959 8 D
cd211a 80.0 1.48 110 176 143 0.15 0.6 12.9 30.6 1.014 0.976 9 D
cd211b 80.0 1.41 110 176 143 0.10 0.2 13.0 26.1 1.017 0.987 8 D
cd212a 80.0 1.50 110 176 143 0.26 3.4 17.2 54.3 1.025 1.043 17 C
cd212b 80.0 1.46 110 176 143 0.25 2.9 16.9 52.3 1.022 1.065 19 C b
cd213a 80.0 1.40 110 176 143 0.33 6.6 " 19.7 71.6 } } 19 C b
482
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cd213b 80.0 1.42 110 176 143 0.35 7.6 19.7 73.5 1.024 0.986 19 C b
cd220a 80.0 1.97 132 195 164 0 0 15.3 41.2 1.015 1.000 7 D
cd220b 80.0 1.96 132 195 164 0 0 " 15.4 41.0 } } 7 D
cd221a 80.0 1.98 132 195 164 0.13 0.5 16.3 49.7 1.016 1.008 7 D#C
cd221b 80.0 1.98 132 195 164 0.12 0.4 16.3 47.9 1.017 1.026 7 D
cd222a 80.0 1.98 132 195 164 0.25 2.8 19.5 70.1 1.022 1.047 15 C
cd222b 80.0 1.98 132 195 164 0.24 2.5 19.5 62.5 1.019 0.934 15 C#D
cd223a 80.0 1.98 132 195 164 0.38 10.0 " 21.7 96.2 } } 15 C b
cd223b 80.0 1.98 132 195 164 0.36 8.4 " 21.7 92.8 } } 15 C#D
cd230a 80.0 2.37 126 187 157 0 0 17.0 55.9 1.017 1.053 5 D
cd230b 80.0 2.38 126 187 157 0 0 17.0 53.5 1.022 1.053 5 D
cd231a 80.0 2.36 126 187 157 0.11 0.2 " 18.0 56.2 } } 5 D
cd231b 80.0 2.37 126 187 157 0.12 0.4 18.1 57.3 1.023 1.052 6 D
cd232a 80.0 2.37 126 187 157 0.25 2.8 21.4 84.8 1.046 1.061 13 C#D
cd232b 80.0 2.45 126 187 157 0.24 2.5 21.1 79.2 1.015 0.992 13 C
cd233a 80.0 2.35 126 187 157 0.33 6.4 22.5 106.5 1.023 1.089 13 C
cd233b 80.0 2.45 126 187 157 0.35 7.9 " 22.6 105.0 } } 13 C
!De"nition after [5]: C"concertina, D"diamond. Global bending: b.
"Electrical system failure in projectile nose. Average force calculated from impact energy and deformed distance.
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483

Fig. 2. Aluminium foam material properties.
denoted by o
f0
(2.7 g/cm3). The foam plateau stress p
f
is herein de"ned as the average stress at
50% strain, that is, energy absorbed at 50% deformation divided by deformed distance.
Some tests were also carried out in order to evaluate the e!ect of dynamic loading conditions on
the stress strain behaviour of the foam. The test apparatus applied for these tests corresponded to
the equipment applied for the dynamic compression tests of the foam "lled extrusions, described in
Section 6.2. The impact velocity of the dynamically tested foam specimens was equal to 15 m/s. As
seen from Fig. 2, no signi"cant e!ect of this impact velocity on the energy absorption behaviour of
aluminium foam can be seen.
The power law, Eq. (1), is important since measurement of the foam density in the components
tested is the only way to accurately assess the strength of the foam "ller. Based upon such pre-test
density measurements, the foam plateau stress of all test specimens were calculated by use of Eq. (1)
and added to Tables 2 and 3.
4.2. Aluminium extrusions
The circular extrusions were made up of aluminium alloy AA6060. Three di!erent values of the
average wall thickness, being h
1
"1.43, h
2
"1.98 and h
3
"2.36 mm were applied in the current
test programme. All extrusions were initially delivered in a T4 condition. In order to test two

di!erent material strengths p
1
and p
2
for each extrusion wall thickness, half of all extrusion test
specimens manufactured were subjected to heat treatment. These specimens were placed in a heat
chamber holding a temperature of 1903C for a period of 70 min. The specimens were allowed to
cool in air after removal from the heat chamber. The resulting condition of these specimens will
herein be referred to as temper T4H. A summary of the extrusion material properties is given in
Fig. 3. In general, the material properties depend upon extrusion wall thickness. Each stress strain
curve has therefore been designated by p
1
(h
1
), etc.
Langseth and Lademo [22] have experimentally assessed the strain rate dependency of the
AA6060 alloy (temper T4 and T6). According to their observations, insigni"cant strain rate e!ects
are present for this alloy.
As described in Section 4.1, the plateau stress p
f
was chosen as a parameter to describe the
material behaviour of the aluminium foam "ller. Similarly, a single parameter describing the plastic
#ow strength of the extrusion wall is of great advantage for simple crash calculations. If the
material is perfect plastic, the knowledge of the plastic #ow stress is a su$cient material parameter.
Several theoretical results exist, summarised in the work by Jones [5], which enables the designer
to predict the average crush load of thin-walled columns. However, they all assume perfect plastic
material behaviour. Application of these design formulas for materials showing signi"cant strain
hardening, as that shown in Fig. 3, is therefore not straightforward. Obviously, a transformation is
needed that gives the designer the correct plastic #ow stress based on a general stress strain curve.
Abramowicz and Wierzbicki [23] refers to the stress resulting from this transformation as the
energy equivalent #ow stress. However, in this study the energy equivalent #ow stress will be
approximated by de"ning a characteristic stress p
0
. The characteristic stress p
0
is calculated as the
average of the initial yield stress p
0.2
and the ultimate stress p
U
. Both Yang and Caldwell [24],
Langseth and Hopperstad [25] and Hanssen et al. [14,16] obtained fair agreement in results
by using the characteristic stress p
0
as an approximation to the energy equivalent #ow stress.
Tables 2 and 3 give the values for p
0
, p
0.2
and p
U
for all components tested herein.
5. Preparation of test specimens
Fig. 4 illustrates the geometry of the test specimens. The delivered aluminium extrusions were
machined to a length of 280 mm. Care was taken in order to obtain parallel end surfaces. After
machining of the specimen ends, an axisymmetric indentation was formed into the top of the
extrusions to act as a trigger for initiation of the deformation modes. A special tool was designed
for this operation allowing the trigger to be formed without causing additional imperfections
outside the trigger area, see Fig. 4 for "nal trigger geometry. The trigger tool was similar in function
to the tool previously applied and described by Hanssen et al. [16] for indentation of square
extrusions. The lower 50 mm of the specimens was clamped during testing, reducing the e!ective
length of the components to 230 mm.
The cylindrical foam "ller was cut from the transversal direction of the supplied foam sheets by
means of a core drill. However, the top trigger prevented the cylindrical foam "ller to be entered as
one piece. Hence 20 mm of the 280 mm long foam specimens were cut o!. These two parts were
then graded and inserted into the extrusion from each end. The grading had roughly the shape of
the trigger and ensured axial contact between the two pieces of foam "ller, see Fig. 4.

Fig. 3. Material properties of extrusion.

Fig. 4. Test specimen geometry.
6. Experimental test set up
6.1. Static tests
The static tests were carried out in a 500 kN Dartec testing machine, showing an accuracy of
$1% of the applied load. The 50 mm lower end of the test specimens was "xed by means of the
same steel-plate-clamping-device as applied for the dynamic tests, see Figs. 4 and 5. A #at steel plate
parallel to the bottom clamping device was mounted to the top hydraulic actuator of the testing
machine in order to ensure a uniform load distribution. The data logging system applied was
running at a constant frequency of 10 Hz, sampling internal force and deformation signals from the
Dartec machine. The load was applied with a constant rate of 40 mm/min. The specimen was
unloaded and the experiment stopped when the force reached a level two times that of the initial
peak load.
6.2. Dynamic tests
The dynamic testing rig is located at the Structural Impact Laboratory (SIMLab), the Depart-
ment of Structural Engineering * NTNU, Fig. 5. This was the same test rig recently applied by
Hanssen et al. [16] to study dynamic axial crushing of foam "lled square extrusions. Additional
information regarding the functionality and system description can be found in Langseth and
Hopperstad [25] and Langseth and Larsen [26]. Brie#y, the dynamic testing rig consists of a top
pressure chamber connected to a vertical accelerator tube. A projectile is released from the top

Fig. 5. Dynamic test rig arrangement.
pressure chamber and "red down the accelerator tube, subsequently hitting the test specimen. By
adjustment of the pressure in the top chamber, the resulting impact velocity can be speci"ed. The
projectile nose is instrumented with strain gauges from which the impacting force as function of
deformed distance can be obtained, see [25] for details. A total projectile mass of 55 kg was used in
the current test set-up. The impact velocity of the projectile is measured by means of a photocell
system located directly above the test specimen. The recorded signals from the projectile were
subjected to a low pass "lter with a cut-o! frequency o! 1.024 kHz. This more or less removes the
high frequency oscillations set up by elastic stress waves in the projectile during impact. A cover
was placed on the top of the specimens in order to ensure a central impact, Fig. 5.

The kinetic energy (EP
kin
"1/2Mv2
0
) of the projectile at impact added to the loss of potential
energy (EP
pot
"Mgd) should approximately equal the calculated maximum plastic energy (E
pl
)
absorbed in the test specimens. Here d is to be regarded as the maximum deformation of the
specimens. In addition, the permanent deformation can easily be measured after the test (d
real
) and
compared to the integrated signal from the strain gauges (d
calc
). Comparison of these measures can
be used as a rough control of the accuracy of the recorded force time signals. As seen from Table 3,
the results are within 5% for most tests. In general, this implies that the results based on the
recorded signals agree on an integrated basis.
An electrical system failure in the projectile nose caused minor problems with some of the
recorded signals. These tests are marked in Table 3, where the average force has been approximated
by dividing the sum of projectile kinetic and potential energy by deformed distance.
7. Visual observations
The axial crushing of non-"lled cylindrical columns produce two distinctive deformation modes,
namely the axisymmetric mode (Concertina) and the non-axisymmetric mode (Diamond). Which
mode occurs is in general dependent upon geometrical and material properties of the actual
column [2,3,5]. A blend between the two modes is also commonly encountered. The concertina
mode occurs for low values of b/h (thick walled columns), whereas the diamond mode occurs for
more slender tubes (high value of b/h). Figs. 6 and 7 show the deformation pattern of the test
specimens resulting from all possible combinations of the model parameters. As seen, the non-"lled
extrusions (o
f
"0) deform in the diamond mode, regardless of loading condition, extrusion wall
thickness h and extrusion material strength p
0
. Details about number of lobes and type can be
found in Tables 2 and 3 for all tests.
Figs. 6 and 7 are most helpful in studying the e!ect of foam "ller on the deformation behaviour of
the columns. Applying the low-density "ller o
1
apparently has no e!ect on the deformation pattern
compared to the non-"lled extrusions. However, a noticeable shift from non-axisymmetric to
axisymmetric deformation occurs when increasing the foam "ller density from o
1
to o
2
. Given an
extrusion, a critical foam density has to exist that determines the shift from diamond to concertina
mode. It is also evident from Figs. 6 and 7 that the change in deformation modes occurred between
the foam densities o
1
and o
2
for all tests. Hence, it is impossible from the current results to draw
any conclusions upon the e!ect of extrusion wall thickness and extrusion strength on the value of
this critical foam density.
Although Figs. 6 and 7 depict the main deformation behaviour of the test specimens as a function
of the model parameters, a few additional observations will be made in the following.
As noted from Figs. 6 and 7, most specimens deforming in the diamond mode have a concertina
lobe in the top. This is obviously due to the in#uence of the axisymmetric trigger on the "rst
deformation lobe.
The diamond mode of the non-"lled and foam "lled specimens of density o
1
showed six corners
per lobe for the majority of the tests. A deviation from this pattern was found in the number of
corners generated in the non-"lled, thick walled tubes. Most common was the 6 corner mode, but
the four corner mode also occurred, see Fig. 8. A clear distinction in the number of corners between
the static and dynamic loading condition was made for the thin walled extrusions h
1
. All statically

Fig. 6. Static tests, all model parameter combinations.

Fig. 7. Dynamic tests, all model parameter combinations.

Fig. 8. Thick walled tubes, variation in number of corners of the diamond mode.
tested non-"lled specimens showed six corners, whereas three out of four dynamically tested
specimens obtained a total of eight corners, Fig. 9. This tendency was also observed for the thin
walled specimens with the lightest foam "ller.
As mentioned above, the axisymmetric trigger in#uenced the generation of the top lobe.
However, the trigger area might start to deform in two di!erent ways, Fig. 10. The most common
pattern was generated when the section above the trigger folded outwards. For some cases though,
the top section of the whole tube might get squeezed into the bottom section. This was observed to
happen also for foam "lled tubes given that the main deformation pattern was of the diamond type.
Finally, some dynamically loaded specimens obtained a mixture of local folding and moderate
global bending. This was not observed in the static tests. The specimens that obtained this mode
are marked in Table 3, and an illustration is given in Fig. 11 for two selected specimens. As seen
from Table 3, the global bending mode was only obtained for some of the components with the
high-density foam "ller o
3
.
8. Experimental results
8.1. Static tests
8.1.1. General behaviour
The e!ect of the trigger on the initial peak load of a non-"lled extrusion is given in Fig. 12. It is
evident that the axisymmetric notch has reduced the maximum load in the early stages of crushing.

Fig. 9. Thin walled tubes, di!erence in diamond mode between static and dynamic tests.
Fig. 10. Characteristic behaviour of trigger.

Fig. 11. Example of global bending obtained for dynamic loading condition.
Fig. 12. E!ect of trigger on initial peak load, quasi-static reference tests.
The typical e!ect of foam "ller on the force}deformation curve is plotted in Fig. 13. An interaction
e!ect apparently exists, so that the total load of the foam "lled extrusion exceeds that of the sum of
non-"lled extrusion and the uniaxial resistance of foam. This suggests that a model for the average
crushing force could be divided into three distinctive parts being (1) average force of non-"lled
column, (2) uniaxial resistance of foam and "nally (3) an interaction e!ect. The next sections will
focus on means for estimating this interaction e!ect. A complete set of force vs. deformation curves
is given in Section 8.2, comprising all possible parameter combinations.

Fig. 13. Illustration of interaction e!ect, quasi-static reference tests.
8.1.2. Average force
Hanssen et al. [16] developed an additive design formula for the average crushing force of square
foam "lled extrusions. It is natural to attempt the same procedure for determination of the
interaction e!ect of foam "lled circular extrusions. Hence, the total crush force F
avg
is modelled as
F
avg
"F0
avg
#
p
4
p
f
b2
i
#C
avg
pa
f
p(1~a)
0
bb
m
h(2~b) (2)
where C
avg
, a and b are dimensionless constants whereas b and h are the outer diameter and wall
thickness of the extrusion respectively. Here, F0
avg
is the contribution from the non-"lled extrusion,
(p/4) p
f
b2
i
is the uniaxial resistance of the foam "ller and the last term describes the interaction
e!ect. The additional parameters de"ned are b
m
"b!h and b
i
"b!2h. In order to "nd an
expression for the interaction e!ect, the dimensionless parameters C
avg
, a and b were chosen so that
the total sum of squares of the residual error between model and experiments was minimised.
Another important aspect is that the procedure for estimating the interaction e!ect should be
carried out by using experimentally obtained values for the average crushing force of non-"lled
extrusions F0
avg
. In this way the determination of the interaction e!ect is independent of any
theoretical model used to describe F0
avg
. Following this procedure, Table 4 shows the result when
minimising the sum of squares with respect to the parameters C
avg
, a and b as function of relative
deformation.
For practical purposes it is convenient to "x a and b and carry out a new optimisation of C
avg
as
function of deformation. Table 4 also shows the results when a"0.5 and b"1.0. Comparing these
cases by regarding the squared Pearson value R2, shows that little loss of accuracy is introduced

Table 4
Interaction e!ect on average crush force
C
avg
pa
f
p(1~a)
0
bb
m
h(2~b)
D
C
% 20 30 40 50 60 S
E
Optimum values of
C
avg
a and b
C
avg
0.58 0.85 0.34 0.28 0.18 0.20
a 0.18 0.29 0.32 0.33 0.36 0.42
b 0.88 1.06 1.38 1.48 1.64 1.74
R2 0.977 0.981 0.983 0.981 0.982 0.907
C
avg
Jp
f
p
0
b
m
h
D
C
(%) 20 30 40 50 60 S
E
a and b "xed C
avg
1.08 2.07 2.45 2.74 2.90 3.76
R2 0.974 0.978 0.981 0.979 0.980 0.906
when "xing a and b. In general, values of R2 close to 1 are found for most cases, which indicates
that the model gives good correlation.
In order for Eq. (2) to be a complete design formula, an expression is needed for the average crush
force F0
avg
of non-"lled extrusions showing diamond mode lobes. Based on the assumption of
perfect plastic material behaviour, Wierzbicki [27] gives the expression
F0
avg
"i
D
p
0
b1@3
m
h5@3 (3)
where i
D
is a dimensionless constant. Ref. [2] gives i
D
"24.87 whereas Ref. [27] states that
i
D
"18.15. Table 5 gives the results in terms of i
D
based upon a correlation between Eq. (3) and
the current experimental results of non-"lled extrusions. As seen, a good approximation, valid for
all relative deformations, can be found by setting i
D
equal to 17.0. The correlation of Eq. (3) with
experimental results is given in Fig. 14.
The total model for the average crush force of foam "lled extrusions can now be written as
F
avg
"17.0p
0
b1@3
m
h5@3#
p
4
p
f
b2
i
#C
avg
Jp
f
p
0
b
m
h (4)
where a corresponding correlation plot for a deformation of 50% is given in Fig. 15. The value of
the interaction constant C
avg
as a function of relative deformation is shown in Fig. 16. As was
pointed out in Ref. [16], the interaction e!ect given by C
avg
increases with deformation. A possible
explanation may be that the foam herein is modelled as a perfect plastic material characterised by
the plateau stress p
f
. In reality, the foam stress level increases as function of deformation (strain
hardening).
As observed in Section 7, a distinctive transformation in deformation behaviour from diamond
to concertina mode occurred for a critical value of the foam density p
fC
. In this case, the expression

Table 5
Investigation of non-"lled extrusions (diamond mode)
F0
avg
"i
D
p
0
b1@3
m
h5@3
D
C
(%) 20 30 40 50 60 S
E
i
D
16.82 16.45 16.80 17.03 17.45 17.91
R2 0.974 0.984 0.992 0.987 0.981 0.982
Fig. 14. Correlation of average force, non-"lled extrusions at D
C
"60%.
used to model the interaction e!ect must also take into account the di!erence in energy absorption
between the diamond and concertina mode. Hence, the interaction e!ect accounts for the change of
mode too. The di!erence in average force of the diamond and concertina mode can be seen by
comparing the theoretical expressions given below.
F0
avg
"i
D
p
0
b1@3
m
h5@3, (Diamond mode, Ref. [27]) (3)
F0
avg
"i
C
p
0
b1@2
m
h3@2, (Concertina mode, Ref. [2]) (5)
For this reason it is important to keep in mind that another expression for the interaction e!ect
would emerge if the non-"lled tubes obtained the concertina mode. However, no experimental data
was obtained for the average force of non-"lled extrusions with concertina mode in the current
investigation (i
C
). This means that the value of the interaction e!ect when concertina modes appear
cannot be determined from the results given herein.

Fig. 15. Correlation of suggested model for, all static tests at D
C
"50%.
Fig. 16. Development of interaction e!ects as function of deformation.

Table 6
Interaction e!ect on maximum crush force
C
.!9
Jp
f
p
0
b
m
h
D
C
(%) 20 30 40 50 60
C
max
1.43 2.03 2.44 2.76 3.24
R2 0.947 0.938 0.933 0.931 0.936
8.1.3. Maximum force
Following the same approach as above and as in Ref [16], the description of the maximum force
F
.!9
of foam "lled extrusions is written as
F
max
"F0
max
#
1
A
Ef
p
4
p
f
b2
i
#C
max
Jp
f
p
0
b
m
h (6)
where A
Ef
is the crush force e$ciency of the foam core. A constant value of A
Ef
equal to 0.85 will
be applied in the following, see also [16,20]. Based upon Eq. (3), the maximum force of non-"lled
extrusions is expressed as
F0
max
"
i
D
A
E0
p
0
b1@3
m
h5@3 (7)
where A
E0
is the crush force e$ciency of the non-"lled, triggered extrusions, see Fig. 1.
As for the average force, optimum values of C
max
and A
E0
were found by correlation with the
maximum forces observed in the experiments. Within the e!ective crush length of non-"lled
extrusions, A
E0
was found to have an average value of 0.58. The results obtained for the interaction
e!ect of foam "lled extrusions represented by C
max
are given in Table 6.
The corresponding development of C
max
as a function of relative deformation D
C
is given in
Fig. 16. Correlation plots for the maximum force are given in Figs. 17 and 18. Contrary to the
average force, the maximum force is strongly trigger dependent which limits the practical use of the
results presented in Table 6.
8.1.4. Stroke ezciency
It was pointed out by Hanssen et al. [16] the complexity involved in determining a model for the
stroke e$ciency S
E
of foam "lled square columns. For circular, non-"lled tubes, Abramowicz and
Jones [2] refers
S0
E
"0.73 (diamond mode) (8)
S0
E
"0.86!0.568Jh/b
m
(concertina mode) (9)
where S0
E
denotes the stroke e$ciency of non-"lled columns.
Introduction of foam "ller in general reduces the e!ective crushing length S
E
of the extrusions.
This is due to two factors, namely (1) the stroke e$ciency of the foam "ller itself (densi"cation or
locking strain) and (2) the altered deformation pattern of the extrusion. For square extrusions [16],

Fig. 17. Model for maximum force, non-"lled extrusions, all static tests for D
C
(S
E
.
Fig. 18. Correlation of design model for maximum force, all static tests at D
C
"50%.

Fig. 19. Modelling the stroke e$ciency of foam "lled extrusions.
the foam "ller generated more lobes in the extrusion walls, which again directly reduces the
e!ective stroke length. The circular foam "lled extrusions add their own complexity based on the
signi"cant shift in deformation pattern from diamond to concertina mode that may occur. The
densi"cation strain e
D
of the foam "ller is given by Gibson and Ashby [21] as
e
D
"1!1.5(o
f
/o
f0
). (10)
The reduction in stroke e$ciency SF
E
of the extrusion wall itself due to the foam "ller was modelled
by Hanssen et al. [16] as
SF
E
"S0
EA1!C
SEA
o
f
o
f0
B
a
B (11)
where a and C
SE
are dimensionless constants. It is important to keep in mind that SF
E
is the
imaginary stroke e$ciency of a non-"lled extrusions with the same deformation pattern as it's foam
"lled counterpart, see Fig. 19.

Fig. 20. Correlation of model for stroke e$ciency S
E
.
Now, both the foam "ller and the extrusion contribute to the total crushing force. It is natural to
assume that if the foam "ller crushing force is small compared to the extrusion, then the stroke
e$ciency of the extrusion would also dominate the stroke e$ciency of the total component, and
vice versa. Hence, the total stroke e$ciency is modelled as a force-weighted average of the foam
in#uenced stroke e$ciency SF
E
and the densi"cation strain of the foam "ller e
D
S
E
"
(F0
avg
#C
avg
Jp
f
p
0
b
m
h)SF
E
#(p/4 p
f
b2
i
)e
D
F
avg
(12)
This model was correlated with the current experimental data. The best "t was obtained when
applying S0
E
"0.76, C
SE
"17 and a"2. Fig. 20 shows the result.
8.2. Dynamic tests
The impact velocities applied in the dynamic tests were determined so that the initial kinetic
energy of the projectile corresponded with the energy absorption of the components at D
C
"S
E
as
observed in the quasi-static tests. It is obvious that this programme does not allow for a systematic
investigation of the e!ect of the impact velocity on the behaviour of the foam "lled components.
Hanssen et al. [16] also followed the same scheme on square foam "lled extrusions, but strongly
bene"ted from complementary experimental and numerical data on non-"lled square extrusions

obtained by Langseth and Hopperstad [25] and Langseth et al. [28]. For the present study, no
such additional data are available on similar circular extrusions. Hence, contrary to Hanssen et al.
[16], which managed to calibrate a simple model based on the available data. This section will
focus on giving a qualitative description of the dynamic e!ects observed in the present investiga-
tion. The dynamically obtained force deformation curves will be displayed and compared with the
corresponding quasi-static tests. Hopefully, this information will be to the bene"t of future
validation of numerical FEM simulations.
Fig. 21 shows the obtained force vs. deformation curves for all combinations of the model
parameters. A general trend is that the dynamically obtained force signals show higher values than
the corresponding quasi-static tests. This can be seen by studying the `energy-ratioa in Fig. 21,
which is the dynamic mean force divided by the static mean force. For most tests, the energy ratio is
a decaying function of deformation. It was pointed out in Section 4 that both the foam "ller and the
extrusion material showed no velocity dependence. Hence it is natural to assume that the observed
dynamic increase in mean load is due to inertia forces arising in the extrusion walls. Langseth and
Hopperstad [25] and Langseth et al. [28] originally gave this explanation based on their
experimental and numerical investigations on square, non-"lled AA6060 aluminium extrusions. In
total, all the observations made in [16,25,28] concluded that the energy ratio of non-"lled
extrusions was an increasing function of impact velocity. Now, studying Fig. 1, going from left to
right, it is apparent that an increase of the impact velocity does not increase the energy ratio in
general. In fact, the trend is for the energy ratio to decrease. However, going from left to right in
Fig. 1 also means an increase in the density of the foam "ller. Since the foam "ller shows no velocity
dependence and contributes to a fair amount of the energy absorption, it is no surprise that the
energy ratio does not increase for the foam "lled specimens compared to their non-"lled counter-
parts.
The error introduced by applying the static design formula in Eq. (4) to predict dynamic average
force is illustrated in Fig. 22 for a relative deformation of 40%. On average, the dynamic crush force
is 14% higher than what the static design formula predicts.
This was also observed by Hanssen et al. [16] for square, foam "lled extrusions. By relating all
dynamic e!ects to the extrusion, they extended Eq. (4) in the following way:
F
avg
"F0D
avg
(v
0
)#
n
4
p
f
b2
i
#C
avg
Jp
f
p
0
b
m
h. (13)
Here, F0D
avg
(v
0
) is the average crush force of non-"lled extrusions, corrected to take into ac-
count dynamic inertia e!ects. This function is not known for the present circular extrusions.
However, an indication whether Eq. (13) works or not can be sought by replacing F0D
avg
(v
0
)
with the experimentally obtained crush load of the dynamically loaded non-"lled specimens.
The impact velocity of the foam "lled specimens is higher than that of the non-"lled extrusions, so
this model should be expected to underestimate the average crush force when high impact velocities
occur. The result is given in Fig. 23. The average error is now signi"cantly reduced. Another
observation is that the average force is underpredicted only for the extrusions showing concertina
modes. These extrusions also had the densest foam "ller and thus the highest value of the impact
velocity. This again indicates that F0D
avg
(v
0
) should be an increasing function with respect to impact
velocity.

Fig. 21. Comparison of static and dynamic force}deformation curves.

Fig. 22. Use of static design formula to predict dynamic average crush force, D
C
"40%.
Fig. 23. Design formula corrected for dynamic loading, D
C
"40%.

9. Conclusions
Experimental tests were carried out where the main intention was to study the e!ect of foam "ller
on the energy absorption characteristics of circular aluminium extrusions. Attention was given to
present an extensive data set consisting of quasi-static as well as dynamic loading conditions.
Together with previously presented data on foam "lled square extrusions, this database provides
valuable information for future validation of computer codes. The main focus of the treatment of
the experimentally obtained data was to develop a design formula to estimate the average crush
force of the components. An additive model where the total crush force was divided into three parts
being (1) average crush force of non-"lled extrusions, (2) uniaxial resistance of foam "ller and "nally
(3) an interaction e!ect, showed acceptable correlation with the experimental data. The dynamic
loading condition was found to increase the mean load compared to the static case. The results
indicate that this load increase was due to inertia e!ects arising in the extrusion walls during
crushing (the velocity dependence of foam "ller and strain rate sensitivity of the extrusion material
was ruled out by material testing). As a result, the foam "lled extrusions in general showed less
dependency of the loading condition than similar non-"lled extrusions.
Acknowledgements
This research programme was made possible by material deliverance and economic support
from Hydro Aluminium a/s. The machining of the test specimens was carried out by Terje Pedersen
and Morten Ringlie Fla
> of Olav Haldorsen's crew. Olav Haldorsen and Morten Ringlie Fla
>
designed and machined the trigger mechanism. Trygve Meltzer, Ole Aunr+nning and Kjell Brevik
gave vital technical assistance in the laboratory.
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