Metal Foams: A Design Guide

166 Metal Foams: A Design Guide
Using the relation vD D cplεD and the momentum balance D D pl C cplvD,
the downstream compressive stress, D, exerted on the impacting mass is
D D
MV
pl C
εD
2 0 2 plAℓεD/⊲1 εD⊳
M C Aℓ/⊲1 εD⊳
⊲11.25⊳
Thus, the compressive stress decelerating the mass M decreases with the length
of foam compacted, ℓ/⊲1
on the mass is
εD⊳. In the limit ℓ D 0 the peak compressive stress
⊲ D⊳peak D pl C V2 0
εD
⊲11.26⊳
in agreement with the findings of the shock wave analysis above. The length of
foam ℓ/⊲1 εD⊳ required to arrest the mass is determined by putting vD D 0
in the above equation, giving
1
ℓ
εD
D MV20 2 plεDA
⊲11.27⊳
On noting that plεD equals the energy, W, absorbed by the foam per
unit volume, we see that the minimum length of foam required for energy
absorption is obtained by selecting a foam with a maximum value of W,
consistent with a value of upstream stress pl which does not overload the
structure to which the foam is attached. The plots of W versus pl for metallic
foams (Figures 11.3, 11.4 and 11.5) are useful for this selection process.
11.6 Blast and projectile protection
Explosives create a pressure wave of approximately triangular profile, known
as a ‘blast’ (Smith and Hetherington, 1994). Protection against blast involves
new features. The blast imparts an impulse, Ji, per unit area of a structure,
equal to the integral of the pressure over time:
�
Ji D pdt ⊲11.28⊳
The blast wave is reflected by a rigid structure, and the details of the
pressure–time transient depend on the orientation of the structure with respect
to the pressure wave. In design, it is conservative to assume that the structure
is at normal incidence and fully reflects the blast. Figure 11.15 shows the
peak pressure, p0, and the resulting impulse, Ji, caused by the detonation of
a charge of TNT, at a radial distance, R, from the charge. Curves are shown
for a reflected blast in air, and in water. The impulse and the distance are
normalized by the cube root of the mass, M, of the charge in kg. As an