Metal Foams: A Design Guide

Energy management: packaging and blast protection 165
contribution to strength is 10% of the static contribution, giving
�
0.1 plεD
Vt D
⊲11.21⊳
Recall that the plateau strength for metal foams is approximated by
(Table 4.2)
pl
D C1
ys
�
s
� 3/2
⊲11.22⊳
where ys is the yield strength of the solid of which the foam is made and C1
is a constant with a value often between 0.2 and 0.3. The densification strain
scales with relative density according to εD D ˛ ˇ⊲ / s⊳, where˛ ³ 0.8
and ˇ ³ 1.75. Hence, the transition speed, Vt, depends upon foam density
according to
�
Vt D 0.1C1 ys
�1/2 �
s
�1/4 �
˛
s
�1/2 ˇ
s
⊲11.23⊳
Examination of this relation suggests that Vt shows a maximum at
/ s D ˛/3ˇ D 0.15. Now insert some typical values. On taking C1 D 0.3,
/ s D 0.15, ys D 200 MPa and s D 2700 kgm 3 ,wefindVtD21.5ms 1
(77 km h 1 ). For most practical applications in ground transportation, the
anticipated impact speeds are much less than this value, and we conclude
that the quasi-static strength suffices at the conceptual design stage.
Kinetic energy absorber
Insight into the optimal design of a foam energy absorber is gained by considering
the one-dimensional problem of end-on impact of a long bar of foam of
cross-sectional area A by a body of mass M with an impact velocity V0, as
sketched in Figure 11.14(b). After impact, a plastic shock wave moves from
the impact end of the bar at a wave speed cpl. Consider the state of stress in
the foam after the plastic wave has travelled a distance ℓ from the impacted
end. Upstream of the shock, the foam is stationary (except for a small speed
due to elastic wave effects) and is subjected to the plateau stress pl. Downstream,
the foam has compacted to a strain of εD, is subjected to a stress D
and moves at a velocity vD equal to that of the mass M. An energy balance
gives
1
2
�
M C Aℓ
1 εD
�
v 2 D C plεDA
ℓ
1 εD
D 1
2 MV2 0
⊲11.24⊳