Metal Foams: A Design Guide

84 Metal Foams: A Design Guide
with the inversion
⎛
1 P
⎜
˛ D 3 ⎝
2
1 C P
⎞
⎟
⎠
1/2
⊲7.14⊳
P It would appear that the measurement of in a uniaxial compression
test offers a quick and simple method for estimation of the value for ˛, and
thereby the shape of the yield surface. Preliminary experience suggests that
the measurement of P is best done by compressing a sample, with suitably
lubricated loading platens, to a uniaxial strain of 20–30%.
Having defined the yield surface shape, it remains to stipulate how the yield
surface evolves with strain. For simplicity, we shall assume that isotropic hardening
occurs: the yield surface grows in a geometrically self-similar manner
with strain; the limited measurements of the yield surface for metallic foams
approximate this behavior (see, for example, Figure 7.3 for the case of Alporas
with an initial relative density of 0.16). Yield surfaces are displayed for the
initial state, and for 10% and 30% uniaxial pre-strain. We note that the yield
surfaces are smooth and geometrically self-similar.
We assume that the strain-hardening rate scales with the uniaxial compression
response as follows. The plastic strain rate is again taken to be normal
to the yield surface (7.11), and specified by the analogue of (7.9), given by
Pε P ij D P Oε ∂8
⊲7.15⊳
∂ ij
Normalized effective stress
2.5
2
1.5
1
0.5
0
0 0.5
Initial
10%
Uniaxial
strain
1 1.5 2 2.5
Normalized mean stress
10%
Volumetric
strain
Figure 7.3 The evolution of the yield surface with strain for an Alporas
foam with an initial relative density of 0.16