Metal Foams: A Design Guide

86 Metal Foams: A Design Guide
the curves almost collapse unto a unique curve for a given material, up to its
peak strength.
The constitutive law for plastic flow is completely specified by the yield
surface as defined by equations (7.11) and (7.12), and by the flow rule (7.15),
with the definitions (7.16b) and (7.18). Explicit expressions can be given for
⊲∂8/∂ ij⊳ in (7.15), and for P O in (7.18). They are:
� �
∂8 1 3 Sij ˛2 m
D �
∂
�
ij ˛�
� C υij 2 2 O 3 O
1 C
3
�
PO
1 3 Sij
D � � Pij C
2 2 O
˛2 m
3 O Pkk
�
�
1 C
� ˛
3
⊲7.19⊳
⊲7.20⊳
An example of the use of these equations is given in Chapter 9, Section 9.5.
7.3 Postscript
Two final, cautionary, comments. Insufficient experimental data are available
to be fully confident about the accuracy of the isotropic hardening law (7.18).
There is some evidence that the rate of hardening for hydrostatic compression
may be greater than that in simple compression. Indeed, experimental measurements
of the hydrostatic and uniaxial compression responses of Alporas and
Duocel suggest that the hardening rate is faster for hydrostatic compression
(see Figure 7.5 for data presented in the form of true stress versus logarithmic
plastic strain curves).
Nominal pressure
(or Uniaxial stress) (MPa)
20
18
16
14
12
Hydrostatic compression
Uniaxial compression
Alulight
30%
10
8
Alporas
6
13%
4
2
Alporas
8%
0
0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8
Nominal (Volumetric or Axial) strain
Figure 7.5 Compression and hydrostatic stress–strain curves for Alporas
and Duocel foams: true stress and logarithmic strain