Biomedical Engineering Reference
In-Depth Information
(a)
(b)
Figure 6.14
Simple idealizations of open-cell structures: (a)
the ball-and-strut model [47]; (b) the cubic strut model [48].
general the dead volume does not need to be entirely covered by strut elements. The
important geometrical parameter in this model is the ratio of the sphere diameter
to the free strut length.
A cubical arrangement of square-section struts (cf. Figure 6.14(b)) was proposed
by Gent and Thomas [48], who investigated the elastic behavior and the tensile
rupture of open-cell foamed elastic materials. In their analysis, they assumed
that the junctions of the struts are essentially undeformable compared with the
struts themselves, and they referred to this as
dead volume
. An equivalent model
for closed-cell systems was proposed by Matonis [50]. Cubic strut models have
been extensively applied by Gibson and Ashby to different types of materials
(bone, polymer, and metal foams [30, 31, 51]) to investigate different macroscopic
properties (e.g., elastic and plastic behavior) by consideration of beam bending
of the struts. Anisotropy can be simply incorporated in this cubical model by
uniformly stretching it in one of the principal directions, [52]. It should be noted
here that the cubic strut model shown in Figure 6.14(b) behaves in the linear elastic
range according to Hooke's law for orthotropic materials with cubic structure, cf.
Section 6.3.3. Nevertheless, the derived material properties are in most of the cases
assigned to an equivalent isotropic material.
For the cubic plate models shown in Figure 6.15, the primary deformation mode
is bending as in the case of the cubic strut model. However, these models were
motivated by the cancellous bone of lower porosity, where the structure transforms
into a more closed network of plates. Some of these plate elements have small
perforations in them resulting in cells that are not entirely closed [53].
Columnar models with a hexagonal cross section are shown in Figure 6.16 for a
rodlike structure (a) and a platelike structure (b) of cancellous bone. These models
are motivated by bones where the loading is mainly uniaxial (e.g., as the vertebrae)
and the trabeculae often develop a columnar structure with cylindrical symmetry
[54, 55]. The elastic behavior of such a model is described by generalized Hooke's
law for transverse isotropic materials, cf. Section 6.3.4.
