Reference: A.P. Roberts and E.J. Garboczi, J. Mech. Phys. Solids
50 (1), 33-55 (2002).
Most cellular solids are random materials, while practically all
theoretical results are for periodic models.
To be able to generate theoretical
results for random models, the finite element method (FEM) was used to study
the elastic properties of open-cell solids. We have computed the
density ( ρ ) and microstructure dependence of the
Young's modulus (
E) and Poisson's ratio (ν)
for four
different isotropic random models. The models were
based on Voronoi tessellations,
level-cut Gaussian random fields, and nearest neighbour node-bond rules.
These models were chosen to broadly represent the structure of
foamed solids and other (non-foamed) cellular materials.
At low densities, the Young's modulus can
be described by the relation
E 
ρ
n. The exponent
n and constant of proportionality depend
on microstructure. We find 1.3 <
n < 3, indicating
a more complex dependence than indicated by periodic cell theories,
which predict
n=2. The observed variance in the exponent was found
to be consistent with experimental data.
We found that the Voronoi tessellation model, which is often used as a
common model of isotropic foamed solids,
exhibits incompressibility
(ν ≈
½)
at
low densities. This behaviour is not observed experimentally. Our studies
showed the result was robust to polydispersity and that a relatively
large number (15 %) of the bonds must be broken to reproduce the
experimental Poisson's ratio.