Metal Foams: A Design Guide

164 Metal Foams: A Design Guide
quickly along the bar at an elastic wave speed cel D p E/ and brings the bar
to a uniform stress of pl and to a negligibly small velocity of v D pl/⊲ cel⊳.
Trailing behind this elastic wave is the more major disturbance of the plastic
shock wave, travelling at a wave speed cpl. Upstream of the plastic shock
front the stress is pl, and the velocity is vU ³ 0. Downstream of the shock
the stress and strain state is given by the point D on the stress–strain curve:
the compressive stress is D, the strain equals the densification strain, εD, and
the foam density has increased to D D /⊲1 εD⊳.
Momentum conservation for the plastic shock wave dictates that the stress
jump ⊲ D pl⊳ across the shock is related to the velocity jump, vD, by
�
D
�
pl D cplvD ⊲11.16⊳
and material continuity implies that the velocity jump, vD, is related to the
strain jump, εD, by
vD D cplεD
⊲11.17⊳
Elimination of vD from the above two relations gives the plastic wave speed,
cpl:
cpl D
�
⊲ D pl⊳
εD
This has the form
cpl D
�
Et
⊲11.18⊳
⊲11.19⊳
where the tangent modulus, Et, is the slope of the dotted line joining the
downstream state to the upstream state (see Figure 11.13), defined by
Et
⊲ D pl⊳
εD
⊲11.20⊳
The location of the point D on the stress–strain curve depends upon the
problem in hand. For the case considered above, the downstream velocity,
vD, is held fixed at the impact velocity, V; then, the wave speed cpl is
cpl vD/εD D V/εD, and the downstream stress D is constant at D pl C
cplvD D pl C V 2 /εD. This equation reveals that the downstream stress, D,
is the sum of the plastic strength of the foam, pl, and the hydrodynamic
term V 2 /εD. A simple criterion for the onset of inertial loading effects in
foams is derived by defining a transition speed Vt for which the hydrodynamic