Metal Foams: A Design Guide

Chapter 7
A constitutive model for metal foams
The plastic response of metal foams differs fundamentally from that of fully
dense metals because foams compact when compressed, and the yield criterion
is dependent on pressure. A constitutive relation characterizing this plastic
response is an essential input for design with foams.
Insufficient data are available at present to be completely confident with
the formulation given below, but it is consistent both with a growing body
of experimental results, and with the traditional development of plasticity
theory. It is based on recent studies by Deshpande and Fleck (1999, 2000),
Miller (1999) and Gibson and co-workers (Gioux et al., 1999, and Gibson and
Ashby, 1997).
7.1 Review of yield behavior of fully dense metals
Fully dense metals deform plastically at constant volume. Because of this,
the yield criterion which characterizes their plastic behavior is independent
of mean stress. If the metal is isotropic (i.e. has the same properties in all
directions) its plastic response is well approximated by the von Mises criterion:
yield occurs when the von Mises effective stress e attains the yield
value Y.
The effective stress, e, is a scalar measure of the deviatoric stress, and is
defined such that it equals the uniaxial stress in a tension or compression test.
On writing the principal stresses as ( I, II, III), e canbeexpressedby
2 2 e D ⊲ I II⊳ 2 C ⊲ II III⊳ 2 C ⊲ III I⊳ 2
⊲7.1⊳
When the stress s is resolved onto arbitrary Cartesian axes Xi, not aligned with
the principal axes of stress, s has three direct components ( 11, 22, 33) and
three shear components ( 12, 23, 31), and can be written as a symmetric 3 ð 3
matrix, with components ij. Then, the mean stress m, which is invariant with
respect to a rotation of axes, is defined by
m
1
3⊲ 11 C 22 C 33⊳ D 1
3 kk ⊲7.2⊳
where the repeated suffix, here and elsewhere, denotes summation from 1 to
3. The stress s can be decomposed additively into its mean component m