A Superposition Procedure for Calculation of Effective Diffusion and Elastic Parameters of Sparsely Porous Materials

Effective material parameters for diffusion and elastic deformation are calculated
for porous materials using a continuum theory-based superposition procedure. The theory that is limited to two-dimensional cases, requires that the pores are sufficiently sparse. The method
leads to simple manual calculations that can be performed by, e.g. hospital staff at clinical
diagnoses of bone deceases that involve increasing levels of porosity. An advantage is that the
result relates to the bone material permeability and stiffness instead of merely pore densities.
The procedure uses precalculated pore shape factors and exact size scaling. The remaining
calculations do not require any knowledge of the underlying field methods that are used to
compute the shape factors. The paper establishes the upper limit for the pore densities that are
sufficiently sparse. A cross section of bovine bone is taken as an example. The superposition
procedure is evaluated against a full scale finite element calculation. The study compares the
pore induced change of the diffusion coefficient and elastic modulus. The predictions differ
between superposition and full scale calculations with 0.3% points when pore contribution
to the diffusion constant is 3–7%, and 0.7% points when the pore contribution to the modulus
of elasticity is 4.5–5%. It is uncertain if the error is in the superposition method, which is
exact for small pore densities, while the full scale finite model is not.