Metal Foams: A Design Guide

ys crushes axially at the load
Energy management: packaging and blast protection 159
Fm D 4 r 1/3 t 5/3 ys ⊲11.8⊳
The load remains roughly constant until the folds of the tube lock up at a
compaction strain ε Tube
D , giving an axial displacement
υ D ℓε Tube
D
The energy absorbed per unit volume of the tube is then
W Tube
v
D Fmυ
r 2 � �5/3 t
D 4
ℓ r
ysε Tube
D
⊲11.9⊳
The quantity 2t/r is the effective ‘relative density’ of the tube, / s, giving
W Tube
v
D 2 1/3
�
s
� 5/3
ysε Tube
D
The foam absorbs an energy per unit volume of
W Foam
v
W Tube
v
W Foam
v
D C1
�
s
� 3/2
ysε Foam
D
⊲11.10⊳
⊲11.11⊳
(using equations (11.4a) and (4.2)) with C1 ³ 0.3. The densification in both
the tube and the foam involves the folding of tube or cell walls until they
touch and lock-up; to a first approximations the strains εTube D and εFoam D are
equal at the same relative density. Thus the tube is more efficient than the
foam by the approximate factor
� �1/6 ³ 4.2
⊲11.12⊳
s
For all realistic values of / s the tube absorber is more efficient than the
foam, on an energy/volume basis, by a factor of about 3.
The equivalent results for energy absorbed per unit weight are
W Tube
w
D Fmυ
2 rtℓ s
D 2
or, replacing 2t/r by / s,
W Tube
w
D 21/3
�
s
� 2/3
� �2/3 t
r
ys
s
ε Tube
D
ys
s
ε Tube
D
⊲11.13⊳