A numerical study of the crushing of circular aluminium tubes with and without aluminium foam fillers has been carried out to investigate their buckling behaviours under axial compression. A crushing mode classification chart has been established for empty tubes. The influence of boundary conditions on crushing mode has also been investigated. The effect of foam filler on the crushing mode of tubes filled with foam was then examined. The predicted results would assist the design of crashworthy tube components with the preferred crushing mode with the maximum energy absorption

Pled, Florent

Wen, Cui'e

Yan, Wenyi

University of Queensland eSpace

5th Australasian Congress on Applied Mechanics, ACAM 2007  
10-12 December 2007, Brisbane, Australia 
Crushing Modes of Aluminium Tubes under Axial Compression 
 
Florent Pled1, Wenyi Yan2 and Cuie Wen 3 
 
1Département de Génie Mécanique, Ecole Normale Supérieure de Cachan, France  
2Department of Mechanical Engineering, Monash University, Clayton, Victoria 3800, Australia 
3Center for Material and Fibre Innovation, Deakin University, Geelong, Victoria 3217, Australia 
 
Abstract: A numerical study of the crushing of circular aluminium tubes with and without aluminium 
foam fillers has been carried out to investigate their buckling behaviours under axial compression. A 
crushing mode classification chart has been established for empty tubes. The influence of boundary 
conditions on crushing mode has also been investigated. The effect of foam filler on the crushing 
mode of tubes filled with foam was then examined. The predicted results would assist the design of 
crashworthy tube components with the preferred crushing mode with the maximum energy absorption. 
Keywords: finite element simulation, porous materials, aluminium foam, crushing mode, crashworthy 
component. 
1 Introduction 
In the design of metallic energy dissipating systems, thin-walled circular aluminium tubes under 
axial loading conditions have been identified for several decades as very efficient impact energy 
absorbing elements, because of their progressive axial folding [1, 2]. When a circular thin-walled tube 
is crushed axially, it collapses either axisymmetrically or non-symmetrically, depending primarily on 
the ratio of the diameter to the thickness ( /D t ), but also on the ratio of the length to the diameter 
( /L D ). The axisymmetrical mode is often known as the ring mode or concertina mode, while the 
non-symmetrical mode is referred to as diamond mode. The diamond mode is characterised by the 
number of lobes. For certain values of /D t  ratio, a tube may start collapse with the ring mode and 
then switch to the diamond mode, hence exhibiting a mixed mode. Tubes can also collapse in Euler-
type buckling mode, which is a catastrophic collapse involving large bending of the tube and resulting 
in considerable loss of energy absorbing capacity. If the crushing modes of circular tubes with given 
geometric dimensions can be accurately predicted, then initiation of the preferred concertina mode 
with the highest energy absorption values can be guaranteed without the need for physical testing [3]. 
Recently, there has been an increasing interest in using aluminium foams as cores inside thin-
walled metallic tubes for energy absorption. Previous study indicates that the energy absorption of the 
foam-filled tubes can be improved significantly compared to empty tubes. Therefore aluminium foam-
filled tubular structures offer significant potential for industry applications where crashworthiness and 
weight are critical due to their unique combination of physical and mechanical properties, and 
excellent energy absorption. 
Before quantifying the energy absorption by an empty tube or a foam-filled tube, it is important to 
understand the crushing modes as different crushing mode will lead to different energy absorption 
capacity. Finite element analyses are carried out in this paper to address this issue.    
2 Numerical Model 
Finite Element models were developed using the commercial package ABAQUS to simulate the 
crushing. A typical FE model consists of a cylinder clamped by two rigid compression plates. Both 4-
node shell and 8-node solid elements are considered to model the empty tube. The foam core was 
modelled by using 8-node solid elements when modelling foam-filled tube.  
  
Table 1  All the boundary conditions involved in the crushing simulation 
The axial compression is 
simulated by moving down the upper 
plate while the lower plate is 
stationary. Several boundary 
conditions reflecting the connection 
between the empty tube and the 
plates are studied. To our 
knowledge, the influence of the 
boundary conditions on crushing 
mode of empty tubes has not been 
paid much attention in literature yet. 
All the boundary conditions studied 
in the crushing simulations are listed 
in Table 1.  
The value of the friction coefficient f between the tube and the foam is chosen as 0.1. The materials 
data are obtained from uniaxial compression tests [4]. ABAQUS contains an extensive library to model 
the behaviour of various engineering materials. It includes models for metal plasticity and crushable 
foam plasticity. 
The tube is an annealed aluminium Alloy A6063T5. Its Youngs modulus tE  and Poissons ratio tn  
are 68.9 GPa and 0.33, respectively. The initial yield stress Yts  is 48 MPa. The density of the 
aluminium tube tr  is 2700 kg/m
3. The tube materia l is assumed to be elastic-plastic with isotropic 
strain hardening, which is described by a power-law work hardening.  The hardening data were 
calibrated from uniaxial compressive tests curves [4].  
The foam is treated as an isotropic homogeneous material. The relative density of the foam, 
defined as the fraction of space occupied by the solid, is determined by the porosity rP  [3]: 
(1 ),f r
a
P
r
r
r
= = -         (1) 
where fr and ar  are the densities of the foam and the aluminium solid, respectively. In the past, 
various analytical, experimental and numerical investigations have been carried out to derive material 
models for metal foams. It was found that the mechanical behaviour of metal foams depends on the 
relative density r  significantly. The Youngs modulus of the foam, fE , can be estimated from the 
relative density r  by [5]: 
2
1
f
a
E
C
E
r= ×          (2) 
where aE  is the Youngs modulus of the aluminium solid. In the previous experimental study [4], the 
aluminium foam samples have closed-cell structure. 1C  is a constant related to the cell geometry of 
the foam, which was calibrated as 0.3 from the experimental results [4]. 
 The solid cell wall material of aluminium have a Youngs modulus aE  of 68.9 GPa, a yield stress 
Yas  of 170 MPa and a density  of 2700 kg/m
3, according to the experimental data. The foam sample 
has a porosity rP  of 91.5%, hence we can estimate the relative density r  which is equal to 0.085 and 
the density of the foam, fr , which is equal to 229.5 kg/m
3. According to Eq. (2), the Youngs modulus 
of the foam, fE , is obtained as 159 MPa. The linear elastic regime in the stress-strain curve of the 
foam sample is characteri zed by an elastic Youngs modulus of 160 Mpa. A value of fn  = 0.33 has 
been chosen in our numerical simulation [6]. 
The crushable foam plasticity model developed by Deshpande and Fleck [7] is applied to simulate 
the plastic behaviour of the aluminium foam. The yield function for this model is defined as: 
 Bottom of the tube Top of the tube 
1st  boundary 
condition 
tie constraint roller 
2nd  boundary 
condition 
Fixed
0
0
ru
uq
=ì
í
=î
 fixed
0
0
ru
uq
=ì
í
=î
 
3rd  boundary 
condition 
Fixed
0
0
ru
uq
=ì
í
=î
 roller 
4th  boundary 
condition 
Roller roller 
  
2 2 2
2 01 ( / 3)
eq m Y
s a s
a
+
- =
+
        (3) 
where eqs  is the Von Mises effective stress, ms  is the mean stress or pressure stress, a  is the 
shape factor of the yield surface and Y  is the hardening material function. The value of a  can be 
determined by the uniaxial and hydrostatic compressive test, which is chosen as 1.58. The plastic 
Poissons ratio pn  = 0.18.  
 
3 Results and Discussion 
3.1 Mode classification chart for empty tubes 
The crushing mode of an empty tube strongly depends on the geometry of the structure. First, the 
numerical simulations are compared with the available experimental data provided in Ref. [3], and 
analyzed with respect to the plastic folding mode. The first type of boundary conditions of Table 1 has 
been applied as it is close to the experimental conditions. 
 
 
Fig. 1  Comparison of crushed shapes of empty tubes (a) shell-element simulation, (b) experiments and (c) solid-                                                        
element simulation 
 
The concertina modes have been correctly predicted. Fig. 1 shows the final crushed shape of an 
empty tube, compared with experimental observation. Both shell elements and solid elements were 
applied to model the tube and can successfully simulate the crushed shape. The shell element tube 
folds more frequently due to the ease with which the elements can rotate. The shell element model 
predicts 15 folds, which is exactly the number of folds observed in experiments, whereas the solid 
element model predicts only 12 folds. 
Secondly, parametric study has been carried out to understand the influence of geometric 
parameters on the buckling modes of empty tubes. The range of geometries used ensured that 
various crush modes were observed. Two hundred and four tubes were considered with varying 
/L D  ratio (from 1 to 12) and /D t  ratio (from 10 to 500). Due to the small thickness of the tubes, 
shell elements were applied in the modelling. The crushed shapes have been successfully simulated 
for all the different crushing modes. 
                               
Fig. 2  Axisymmetric concertina mode and Cut view of              Fig. 3  Diamond 3 lobe mode and Bottom view 
    concertina mode            of diamond mode 
  
   
  Fig. 4  Mixed mode and Euler-type buckling mode            Fig. 5  Mode classification chart for aluminium tubes          
The results shown in Fig. 5 are consistent with the previous works. Our chart covers a sufficiently 
large range of /D t  and /L D . For short and thick tubes, concertina mode appears. When the 
thickness of the tube is reduced and the length of the tube is increased, mixed mode occurs and then 
diamond mode is present. For long and very thin tubes, Euler-type buckling mode takes place.  
3.2 Influence of boundary conditions on buckling modes for empty tubes 
Four empty tubes have been selected to investigate the influence of the boundary conditions on 
crushing modes. Under the first type of boundary conditions, four different crushing modes are 
expected from these four representative tubes, see Table 2.  
Table 2  Dimension parameters of aluminium tubes 
Tube 
Length 
L  (mm) 
Outer Diameter 
D  (mm) 
Wall thickness  
t  (mm) /L D  /D t  
Expected crush 
modes 
1 150 50 1.67 3 30 Concertina 
2 200 25 0.28 8 90 Diamond 
3 200 40 0.44 5 90 Mixed 
4 210 30 0.06 7 500 Euler 
 
For the tube 1, concertina (axisymmetric) mode was observed for all the boundary conditions in the 
simulation. Nevertheless for the two symmetric boundary conditions, (2) and (4), the folds appear at 
both sides of the tube, as shown in Fig. 6. 
 
Fig. 6  Comparison of crushed shapes of Tube 1 for different boundary conditions (1) tie  constraint-roller, (2)   
fixed-fixed (3) fixed-roller and (4) roller-roller 
 
For the Tube 2, diamond mode were observed for the boundary conditions (1) and (4) in the 
simulation, whereas a mixed mode was observed for the boundary conditions (2) and (3), as shown in 
Fig. 7. A refinement of the mesh has been done to see the influence of boundary conditions for the 
Tube 2. We observed that mixed modes appeared for the boundary conditions (1), (3) and (4) and 
concertina mode appeared for the boundary condition (2). Therefore the boundary condition (2) seems 
to be the best for obtaining axisymmetric folds and absorbing a higher amount of energy. 
                                             
Fig. 7  Comparison of crushed shapes of Tube 2 for different boundary conditions (1) tie  constraint-roller, (2)   
fixed-fixed (3) fixed-roller and (4) roller-roller 
  
 
For the Tube 3, mixed mode was observed for boundary conditions (1), (2) and (4), but concertina 
mode was observed for the boundary condition (3) in the simulation, as shown in Fig. 8. Therefore 
when one end of the tube is fixed, a concertina mode appears instead of a mixed mode. 
 
Fig. 8  Comparison of crushed shapes of Tube 3 for different boundary conditions (1) tie  constraint-roller, (2)   
fixed-fixed (3) fixed-roller and (4) roller-roller 
 
For the Tube 4, Euler buckling (catastrophic collapse) crush mode was observed for all boundary 
conditions in the simulation. When one end of the tube is fixed or tied to a plate, this part of the tube 
stay in contact with the plate and when a roller boundary  condition is applied at a end of the tube, this 
end of the tube slides over the plate, as shown in Fig. 9. 
 
Fig. 9  Comparison of crushed shapes of Tube 4 for different boundary conditions (1) tie  constraint-roller, (2)   
fixed-fixed (3) fixed-roller and (4) roller-roller 
3.3 Influence of foam fillers on the crushing mode 
According to experiments, an interaction effect usually exists between the tube wall and the foam 
core. The deformation behaviour of aluminium foams provides a resistance to the inwards deformation 
of the tube wall. Hence it causes an outwards expanding of the progressive buckling of tube wall, or a 
shift of deformation mode from diamond to concertina with foam-filling. Numerical simulations of the 
Tubes 1, 2 and 3 filled with aluminium foam have been carried out under the first set of boundary 
conditions. The numerical results carried out in this part are of great importance since they offer the 
possibility to predict the potential effects of filling on the buckling modes of a tube. 
The effect of foam-filling for Tube 1 has two consequences: Foam-filling did not change the 
deformation mode, but it causes a decrease in the plastic fold length. Indeed the presence of 
aluminium foam results in shorter folding length in experiments [4]. Moreover, foam filling also caused 
the progressive buckling expanding outwards noticeably compared to the tube without foam, as shown 
in Fig. 10. 
            
            (a)                       (b) 
Fig. 10  Comparison of crushed shapes of the empty Tube 1 and the foam-filled tube, showing the same 
crushing mode and a decrease in the plastic fold length (a), but an outwards expansion of the buckling (b). 
 
Foam-filling changed the deformation mode from diamond mode to Euler-type buckling mode and 
concertina mode for the Tube 2 and the Tube 3, respectively, see Fig. 11. 
The numerical model seems to capture quite correctly the contribution of the foam for the 
concertina mode. Indeed, the modelled deformed shape of Tube 1 with foam matches the 
experimental deformed shape. The numerical results of Tube 3 show good agreement with the 
experimental observation, as the progressive buckling mode changed from non-symmetric mixed 
mode into concertina mode. The numerical results demonstrate the deformation shift of Tube 2 from 
diamond mode to Euler-type buckling mode due to foam filling. In the reported experimental work [4], 
  
foam filler of the tube converts the diamond mode into concertina mode, but the geometry of the tube 
is different from that of Tube 2. Furthermore the foam used is light (the porosity of the foam core was 
91.5 %) compared to that used in the experiments (90 % of porosity). Therefore it could be interesting 
to change the foam properties and to see the influence on the crushing mode of diamond. The 
interaction effect between the tube and foam might be affected by not only the tube dimension 
parameters, but also the properties of the foam. As foam has been defined as an isotropic 
homogeneous material, only the effect of changing the density of foam has been investigated. The 
intention was to know if the shift from diamond mode to concertina mode due to foam filling mode 
could occur for a tube filled with a foam of higher density. The porosity of the foam has been changed 
from 91.5% to 88%. Finally the Tube 2 filled with a denser foam collapses in Euler-type buckling mode 
and not concertina mode. Further systematic work is required to completely understand the influence 
of foam filler on the crushing mode.  
 
                             
            Tube 2      Tube 3    
Fig. 11  Crushed shapes of the Tube 2 and the Tube 3 filled with a foam of 91.5 % of porosity, showing that 
progressive crushing mode changes from non-symmetric mode into Euler-type buckling mode and concertina 
mode for the Tube 2 and Tube 3, respectively 
 
 
4 Conclusions 
Numerical simulations of the crushing of aluminium tubes with and without foam filler have been 
carried out. The experimental observations of the post-buckling deformation shape have been 
successfully predicted. The influence of geometric parameters on buckling modes of empty tubes has 
been investigated and a mode classification chart has been established. It shows a quite good 
agreement with previous experimental charts. Different boundary conditions for the tube have been 
studied. It shows that when at least one end is fixed is the best set of boundary conditions for energy 
absorption because it creates more axisymmetric folds, whereas when the two ends of the tube are 
free is the worst set of boundary conditions for energy absorption. Our numerical observations give 
excellent results compared to the experiments for the concertina mode. Numerical results for the 
diamond mode show that Euler-type buckling mode could occur for foam-filled tube. Indeed 
progressive crushing mode did not change from diamond mode to axisymmetric concertina mode 
numerically, but aluminium foam converted the diamond mode into the Euler-type buckling mode, 
even with a foam of higher density. Therefore for thick and short tubes, foam-filling seems to be 
effective in enhancing the energy-absorption performance, whereas if a tube is too thin, it may have a 
strong tendency to undergo Euler-type buckling, greatly reducing the energy absorption. 
References 
[1]  N.Jones: Structural impact, (Cambridge University Press, Cambridge, 1989) pp. 1-575.   
[2]  G. Lu and T. Yu: Energy absorption of structures and materials  (Woodhead Publishing Limited, Cambridge, 
UK, 2003) pp. 1-424. 
[3]  W. Yan, E. Durif, Y. Yamada and C. Wen: Mater. Trans. 48 (2007) 1901-1906.   
[4]  Y. Yamada, T. Banno, Z. Xie and C. Wen: Mater. Trans. 46 (2005) 2633-2636. 
[5]  N. Fleck: Cellular Metals and Polymers, ed. By R. F. Singer, C. Koerner, V. Altsaedt and H. Muenstedt 
(Trans Tech Publications, Zuerich, Switzerland, 2005) pp. 3-6. 
[6]  E. Andrews, W. Sanders and L. J. Gibson: Materials Science and Engineering A270 (1999) 113-124.   
[7]  V. S. Deshpande and N. Fleck: Journal of Mechanics and Physics Solids 48 (2000) 1253-1283. 


Crushing Modes of Aluminium Tubes under Axial Compression

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