And Joule heating can be considered by adding the following term into the
general stiffness matrix:
K
Tϕ
nþ1
h
i
¼ À
Z
V
N
T
Á
∂Q
J
nþ1
∂ϕ nþ1
Á NdV ¼¼
Z
V
N
T
Á J À
1
3
ΔJ
Á
∂N
∂x
dV
ð8:64Þ
8.6 Electrical Conduction Equations
The electrical field in a conducting material is governed by Maxwell’s equation of
conservation of charge. Assuming steady-state direct current and no internal volumetric current source, Maxwell equation reduces to
Z
Ω
j Á n dΩ ¼ 0
ð8:65Þ
where:
Ω—is the surface of a control volume.
n—is the outward normal to surface.
j—is the electrical current density defined in Eq. (8.49).
The divergence theorem can be used to convert the surface integral into a volume
integral which yields to:
Z
V
∂
∂x
Á jdV ¼ 0
ð8:66Þ
The equivalent weak form is obtained by introducing an arbitrary, variation,
electrical potential field, δϕ, and integrating over the volume:
Z
V
δϕ Á
∂
∂x
Á jdV ¼ 0
ð8:67Þ
Using first the chain rule and then the divergence theorem, this statement can be
rewritten as
Z
V
∂δϕ
∂x
Á jdV ¼ 0
ð8:68Þ
By introducing the definition of Eqs. (8.48) and (8.49), the governing conservation of charge equation becomes
8.6 Electrical Conduction Equations
415