Biomedical Engineering Reference
In-Depth Information
wave propagation in a long cortical piezoelectric bone with arbitrary cross
section. El-Naggar and Abd-Alla [28] and Ahmed and Abd-Alla [29] further
obtained an analytical solution for wave propagation in long cylindrical
bones with and without a cavity. Silva et al. [30] explored the physicochemical,
dielectric, and piezoelectric properties of anionic collagen and collagen-
hydroxyapatite composites. Recently, Qin and Ye [19] and Qin, Qu, and
Ye [9] presented a thermoelectroelastic solution for internal and surface bone
remodeling, respectively. Later, Qu and Qin [11] and Qu, Qin, and Kang [14]
extended the results of Qin et al. [9] and Qin and Ye [19] to include magnetic
effects. Accounts of most of the developments in this area can also be found
in references 31 and 32. In this chapter, however, we restrict our discussion to
the findings presented in references 11, 19, and 31.
3.2 Linear Theory of Thermoelectroelastic Bone
Consider a hollow circular cylinder composed of linearly thermopiezoelec-
tric bone material subjected to axisymmetric loading. The axial, circumfer-
ential, and normal to the middle surface coordinate length parameters are
denoted by
z,
θ, and
r,
respectively. In the case of thermoelectroelastic bone,
the constitutive equations (2.11) now become [33]
σ= ε+ ε+ε− −λ
c
c
c
e
ET
rr
11
rr
12
θθ
13
zz
31
z
1
σ=ε+ ε+ε− −λ
c
c
c
e
ET
θθ
12
rr
11
θθ
13
zz
31
z
1
σ=ε+ ε+ε− −λ
c
c
c
e
ET
(3.1)
zz
13
rr
13
θθ
33
zz
33
z
3
σ= ε−
c
e
EDe
,
=ε+κ
E
zr
44
zr
15
r
r
15
zr
1
r
Dd
=ε+ε +ε+κ −χ
(
)
e
E
T
z
1
rr
θθ
33
zz
3
z
3
hkHh
=
,
=
kH
r
r
r
z
zz
where
σ
ij
and ε
ij
again represent components of stress and strain
D
i
and
E
i
denote components of electric displacement and electric field
intensity, respectively
c
ij
is elastic stiffness
e
ij
is a piezoelectric constant
κ
i
is the dielectric permittivity
T
denotes temperature change
χ
3
is a pyroelectric constant
λ
i
is a stress-temperature coefficient
