Metal Foams: A Design Guide

82 Metal Foams: A Design Guide
(This prescription for the plastic strain rate enforces it to be incompressible,
that is, Pε P m D 0.) The hardening rate is specified upon assuming that the effective
strain rate Pεe scales with the effective stress rate Pe according to
Pεe Pe/h ⊲7.10⊳
where the hardening modulus, h, is the slope of the uniaxial stress versus
plastic strain curve at a uniaxial stress of level e. In all the above, true
measures of stress and strain are assumed.
7.2 Yield behavior of metallic foams
We can modify the above theory in a straightforward manner to account for
the effect of porosity on the yield criterion and strain-hardening law for a
metallic foam. We shall assume the elastic response of the foam is given
by that of an isotropic solid, with Young’s modulus E and Poission’s ratio
. Since foams can yield under hydrostatic loading in addition to deviatoric
loading, we modify the yield criterion (7.8) to
8 O Y � 0 ⊲7.11⊳
wherewedefinetheequivalent stress O by
O 2 1
⊲1 C ⊲˛/3⊳ 2 ⊳ [ 2 e C ˛2 2 m ] ⊲7.12⊳
This definition produces a yield surface of elliptical shape in ⊲ m e⊳ space,
with a uniaxial yield strength (in tension and in compression) of Y, anda
hydrostatic strength of
�
⊲1 C ⊲˛/3⊳
j mj D
2
Y
˛
The parameter ˛ defines the aspect ratio of the ellipse: in the limit ˛ D
0, O reduces to e and a J2 flow theory solid is recovered. Two material
properties are now involved instead of one: the uniaxial yield strength, Y,
and the pressure-sensitivity coefficient, ˛. The property Y is measured by a
simple compression test, which can also be used to measure ˛ in the way
described below.
The yield surfaces for Alporas and Duocel for compressive stress states
are shown in Figure 7.1. The data have been normalized by the uniaxial
compressive yield strength, so that e D 1andm D 1
3 for the case of uniaxial
compression. We note that the aspect ratio ˛ of the ellipse lies in the range
1.35 to 2.08. The effect of yield surface shape is reflected in the measured