Biomedical Engineering Reference
In-Depth Information
with the subscripts
A
and
T
for the axial and transverse directions,
respectively;
E
A
and
E
T
for the Young's moduli;
G
A
and
G
T
for the moduli of
rigidity; and μ
A
and μ
T
for the Poisson's ratios. The inverse of Equation (2.10)
gives the stiffness equations as
σ=λ+ ε+λε +λεσ=ε
(
2)
G
,
2
G
,
rr
2
T r
2
θθ
1
zz
r
θ
Tr
θ
σ=λ+ ε+λε +λεσ=ε
(
2)
G
,
2
G
,
(2.11)
θθ
2
T
θθ
2
rr
1
zz
θ
z
Az
θ
σ=λε+ε +λ+ε σ= ε
(
)
(
2),
G
2
G
zz
1
rr
θθ
1
1
zz
rz
A z
where
µ
−µ −µ
EE
E
AAT
λ=
,
1
2
(1
)
2
E
TA
AT
22
µ + µ
−µ −µ +µ
EE
E
TAT
AT
λ=
)
,
(2.12)
2
2
(1
)
E
2
E
(1
TA
AT
T
2
(1
−µ −µ
−µ −µ
)
E
E E
TA
AAT
G
=
1
2
(1
)
E
2
E
TA
AT
The strain-displacement relations in cylindrical coordinates are
ε=
∂
∂
u
r
ε=
∂
∂θ
+ε =
∂
v
u
r
w
z
,
,
,
rr
θθ
zz
r
∂
ε=
∂
1
2
∂θ
+
∂
u
v
r
v
r
ε=
∂
∂
1
2
u
z
+
∂
∂
w
r
−
,
,
(2.13)
r
θ
rz
r
∂
ε=
∂
1
2
∂θ
+
∂
w
v
z
θ
z
r
∂
where
u,
v,
and
w
are the components of the displacement vector in the radial,
tangential, and axial directions, respectively. In the absence of body forces,
the corresponding equilibrium equations are
∂σ
∂
1
∂σ
∂θ
+
∂σ
∂
+
σ−σ
rr
r
θ
rz
rr
θθ
+
=
0,
r
r
z
r
∂σ
∂
1
∂σ
∂θ
+
∂σ
2
r
θ
θθ
θ
z
+
∂
+σ=
0,
(2.14)
r
θ
r
r
zr
∂σ
∂
1
∂σ
∂θ
+
∂σ
1
rz
θ
z
zz
+
∂
+σ=
0
rz
r
r
zr
