Biomedical Engineering Reference
In-Depth Information
Table 3.2
Calculated values GPa for a bonelike material
(
V
F
=
0
.
5) using different composites models (Eqs. 3.6, 3.7,
3.9, 3.10, and 3.15).
E
=
0
.
1GPa,
E
=
1GPa,
E
=
10 GPa,
M
M
M
E
F
=
150 GPa
E
F
=
150 GPa
E
F
=
150 GPa
Voigt-Reuss lower,
upper
0.2, 75.1
2.0, 75.5
18.8, 80
Hashin-Shtrikman
lower, upper
0.3, 50
3.0, 50.8
25.7, 58.7
Eq. (3.15), AR
=
1
0.05
0.55
5.5
Eq. (3.15), AR
=
10
0.52
5.0
34.1
Eq. (3.15), AR
=
20
1.9
15.7
58.8
Eq. (3.15), AR
=
large
75.0
75.5
80
small aspect ratios but approaches the upper bound at very large aspect ratios
(Table 3.2).
3.4
Bone as a Composite: Macroscopic Effects
For a first approximation, mineralized tissues are considered as a two-phase
composite of mineral and nonmineral phases, building on the work of Katz [5]. The
information necessary for examination of mineralized tissues within this simple
composite materials framework is twofold: knowledge of the elastic properties of
the individual phases, as discussed above, and detailed knowledge of the relative
proportions of different phases (e.g., their phase fractions) present in the composite
material.
Compositions of mineralized tissues are frequently reported in terms of weight
(Table 3.3). From the weight fraction of mineral, a volume fraction for the mineral
and nonmineral phases can be calculated (Eq. 3.4). These estimates can then be
compared with estimates made on the basis of the material densities (
ρ
i
)usinga
rule-of-mixtures approach (Eq. 3.3). In both cases, the density of the apatite mineral
phase is taken to be 3.1 g cm
−
3
[74] and the density of the remainder (organic phase
and water) is assumed to be unity.
Table 3.3
Mineralized tissue mean composition by weight or by mass density.
Organic Mass density (g cm
-3
)
% weight [75] Water Mineral
Bone
5-10
75
15-20
1.8-2.0
Enamel
2
97
1
2.97
Dentin
10
70
20
2.14
