Biomedical Engineering Reference
In-Depth Information
h i is heat flow
H i is heat intensity
k i is heat conduction coefficient
The associated strain-displacement relations are again defined by
Equation  (2.13), and the related electric fields and heat intensities are
respectively related to electric potential φ and temperature change T as
E r = −φ , r E z = −φ , z , H r = − T , r , H z = − T , z
(3.2)
For quasistationary behavior, in the absence of a heat source, free electric
charge, and body forces, the set of equations for the thermopiezoelectric
theory of bone is completed by adding the following equations of equilibrium
for heat flow, stress, and electric displacements to Equations (3.1) and (3.2):
∂σ
+ ∂σ
+ σ−σ
∂σ
+ ∂σ
+ σ
rr
zr
rr
θθ
zr
zz
zr
=
0,
=
0,
r
z
r
r
z
r
(3.3)
D
r
+
D
z
D
r
h
r
+
h
z
h
r
r
z
r
r
z
r
+=
0,
+=
0,
where Equation (2.14) is rewritten here as the first two terms of Equation (3.3)
for readers' convenience.
The bone remodeling rate equation (2.7) is modified by adding some addi-
tional terms related to electric fields as
=
r E
z E
ε
ε
ε
eAeAeE
()
+
()
+
AeEA
()
+ ε+ε+ε+ ε
(
)
AA
zr z
(3.4)
r
z
rr r
θθ
zz z
where
ij are material constants dependent upon the volume
function e. Equations (3.1)-(3.4) form the basic set of equations for the adaptive
theory of internal piezoelectric bone remodeling.
Ae
E
()and( )
Ae
ε
i
3.3 Analytical Solution of a Homogeneous
Hollow Circular Cylindrical Bone
We now consider a hollow circular cylinder of bone subjected to an external
temperature change T 0 , a quasistatic axial pressure load P, an external
pressure p, and an electric load φ a (o r/a n d φ b ) as shown in Figure  3.1. The
boundary conditions are
T
=
0,
σ =σ =σ =
0,
ϕ =ϕ
at
r
=
a
rr
r
θ
rz
a
(3.5)
TT
=
,
σ =− σ=σ= ϕ=ϕ
p
,
0,
,
t
r
=
b
0
rr
r
θ
rz
b
Search WWH ::




Custom Search