Metal Foams: A Design Guide

A constitutive model for metal foams 85
where the equivalent strain rate P Oε is the work conjugate of O, such that
O P Oε D ij Pε P ij ⊲7.16a⊳
and can be written explicitly as
POε 2
⊲1 C ⊲˛/3⊳ 2 �
⊳ Pε 2 �
1
e C PεP2 2 m
⊲7.16b⊳
˛
A unique hardening relation is given by the evolution of O with Oε. Unless otherwise
stated, the uniaxial compressive stress–strain response is used to define
the O Oε relation, as follows. The true stress versus logarithmic plastic
strain εP curve in uniaxial compression is written in the incremental form
Pε P DP/h⊲⊳ ⊲7.17⊳
where the slope h evolves with increasing stress level . Recall that, for the
case of uniaxial compression (or tension), the above definitions of O and of
POε have been so normalized that O is the uniaxial stress and P Oε is the uniaxial
plastic strain rate. The hardening law (7.17) for uniaxial loading can then be
rewritten as
POε D P O/h⊲O⊳ ⊲7.18⊳
It is assumed that this relation holds also for general multi-axial loading.
Some checks on the accuracy of this approach are given in Figure 7.4: the
measured tensile, compressive and shear stress–strain curves for Alporas and
Al 6101-T6 Duocel foams are shown in terms of O versus Oε. It is noted that
Pressure or true uniaxial stress (MPa)
30
25
20
15
10
5
Hydrostatic compression
Uniaxial compression
Alulight
30%
Alporas
13%
Alporas
8%
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
True (Volumetric or Axial) strain
Figure 7.4 Demonstration of the ability of the equivalent stress O and the
equivalent strain Oε to define uniquely the stress–strain response of Alporas
( D 0.16) and Duocel ( D 0.071) foams. The tension, compression and
shear response are plotted in terms of O and Oε