c 33 ¼ 1=s
E
33 , e 33 ¼ d 33 =s
E
33 ,
ε 33 ¼ ε
T
33 À d
2
33 =s
E
33 , λ 33 ¼ α 33 =s
E
33 ,
p 3 ¼ p 3 À d 33 α 33 =s
E
33 :
ð7:11Þ
The one-dimensional constitutive relations for the currents are
J
p
3 ffi qp 0 μ
p
33 E 3 À qD
p
33 Δp
ð Þ ,3 ,
J
n
3 ffi qn 0 μ
n
3 E 3 þ qD
n
33 Δn
ð Þ ,3 :
ð7:12Þ
We consider static extension. From Eqs. (7.1), (7.2), and (7.3), the one-dimensional
field equations are
T 33,3 ¼ 0,
D 3,3 ¼ q Δp À Δn
ð
Þ ,
ð7:13Þ
J
p
3,3 ¼ 0,
J
n
3,3 ¼ 0:
ð7:14Þ
The substitution of Eqs. (7.10) and (7.12) into Eqs. (7.13) and (7.14) gives four
second-order equations for u 3 , φ, Δp, and Δn. For an electromechanically isolated
rod, the boundary conditions are
T 33 ÆL
ð Þ ¼ 0, D 3 ÆL
ð Þ ¼ 0,
J
n
ÆL
ð Þ ¼ 0, J
p
ÆL
ð Þ ¼ 0:
ð7:15Þ
The four equations for u 3 , φ, Δp, and Δn are linear ordinary differential equations
with constant coefficient. Their general solution can be obtained in a straightforward
manner as
u 3 ¼
e 33
c 33
q
k
2
b ε 33
A 1 sinh k x 3 À L
ð
Þ
½
ŠþA 2 sinh k x 3 þ L
ð
Þ
½
Š
f
g
þA 5 x 3 þ A 6 ,
φ ¼ À
q
k
2
b ε 33
A 1 sinh k x 3 À L
ð
Þ
½
ŠþA 2 sinh k x 3 þ L
ð
Þ
½
Š
f
g
þA 3 x 3 þ A 4 ,
ð7:16Þ
Δp ¼ p 0
q
k B Θ 0
q
k
2
b ε 33
A 1 sinh k x 3 À L
ð
Þ
½
ŠþA 2 sinh k x 3 þ L
ð
Þ
½
Š
f
g
þA 7 x 3 þ A 8 ,
Δn ¼ Àn 0
q
k B Θ 0
q
k
2
b ε 33
A 1 sinh k x 3 À L
ð
Þ
½
ŠþA 2 sinh k x 3 þ L
ð
Þ
½
Š
f
g
þA 7 x 3 þ A 8 ,
ð7:17Þ
7.1 Extension of Homogeneous Rods
179