Biomedical Engineering Reference
In-Depth Information
2
22
1
2
πλ+ + λ
π− λ+ λ+ −λ
[
b
(
2)
G
PPr
]
abP
Gb ar
1
1
1
1
0
u
=−
−
,
r
2
2
2
2
2(
ba
)[(
G
)(
2)
G
]2 (
−
)
2
T
1
1
1
T
(2.64)
2
πλ +λ+
π− λ+ λ+ −λ
[
bP
(
GPz
)]
11
2
T
0
u
=
z
2
2
2
(
ba
)[(
G
)(
2)
G
]
2
T
1
1
1
The strains and stresses can be easily found by substituting Equation (2.63)
into Equations (2.61) and (2.62), respectively.
2.4.2 Extension to Poroelastic Bone with Fluid
Papathanasopoulou et al. [29] extended the theory of internal bone remodel-
ing described in Section 2.4.1 to the case of a porous elastic deformable solid
in the pores of which a viscous compressible fluid flows. They considered a
hollow circular cylinder of poroelastic bone subjected to a quasistatic axial
load -
P
(
t
) and an internal radial pressure
p
(
t
) (see Figure 2.2). The boundary
conditions at the inner and outer surfaces of the cylinder are
σ
rr
+ σ = −
P
(
t
),
σ
r
θ
= σ
rz
= 0
at the inner surface
r
=
a
(2.65)
σ
rr
+ σ = σ
r
θ
= σ
rz
= 0
at the outer surface
r
=
b
(2.66)
The boundary condition at a transverse cross section
S
of the hollow
cylinder is written as
∫
σ+σ=−
(
)
ds
Pt
()
(2.67)
zz
s
Further, making use of Equations (2.6) and (2.51), the constitutive relations
describing the isotropic poroelastic adaptive bone can be written as
P
b
a
p
(
t
)
P
FIGURE 2.2
The poroelastic hollow cylinder and its loading conditions.
