Biomedical Engineering Reference
In-Depth Information
4.6 Extension to Thermomagnetoelectroelastic Solid
In the case of thermomagnetoelectroelastic solids, the constitutive equation
is defined by Equation (3.71) and the remodeling rate equation (4.1) must be
augmented by some additional terms related to magnetic field as follows:
0
0
0
UC
=
(, )
QQ QCEQ
ε
(
)
−ε
(
)
+
(
)
−
EQ
(
)
+
GHQHQ
(
)
−
(
)
ij
ij
ij
i
i
i
i
i
i
(4.73)
=ε+ε+ε+ε+
CC C
C
CE
+
CE
+
GH
+
GH
−
C
rr
rr
θθ θθ
zz
zz
rz
rz
rr
zz
rr
zz
0
where
G
i
(
i
=
r, z
) are surface remodeling coefficients, and
0
0
0
0
0
0
0
0
(4.74)
CC
=ε+ε+ε+ε+
C
C
C
ECEGHGH
+
+
+
0
rr
rr
zz
zz
θθ θθ
rz
rz
rr
zz
rr
zz
We now consider again a hollow circular cylinder of bone subjected to an
external temperature change
T
0
, a quasistatic axial load
P,
an external pres-
sure
p,
an electric potential load φ
a
(o r/a n d φ
b
), and a magnetic potential load
ψ
a
(a n d/o r ψ
b
). The boundary conditions are defined by Equations (3.6) and
(3.77). The governing equations are defined by Equations (3.7) and (3.74)-(3.76).
Then, the solutions of
U
e
and
U
p
to the preceding problem have the same form
as those of Equations (4.2) and (4.3), except that the coefficients
NN
e
and
4
are
p
4
replaced by
e
c
ϕ−ϕ−
α
ψ−ψ
(
)
(
)
15
15
e
e
e
(4.75)
N
=−
CC
+
CG
+
4
zr
r
b
a
zr
r
b
a
c
44
44
e
c
ϕ−ϕ−
α
ψ−ψ
(
)
(
)
p
15
p
15
p
N
=−
CC
+
CG
+
(4.76)
zr
r
b
a
zr
r
b
a
4
c
44
44
Therefore, the solutions to
a
and
b
have the same form as those described in
Section 4.2.4.
As numerical illustration of the analytical solutions described in this
section, we consider again a femur as discussed in Section 3.7, except that
the volume fraction change
e
is now taken to be zero. In addition, the sur-
face remodeling rate coefficients given in Section 4.5 are used. The surface
remodeling rate coefficient for magnetic field is
G
r
= 10
−10
m
2
/Ampere · sec
We distinguish three loading cases:
• Case 1:
T
0
(
t
) = −0.5°C, −0.2°C, 0°C, 0.2°C, 0.5°C, φ
b
− φ
a
= 30V, ψ
b
− ψ
a
= 1A,
p
(
t
) = 1 MPa,
P
(
t
) = 1500 N
