That is included in the internal potential energy, which refers to many forms of the
energy stored in the lattice.
For arbitrary values of W input and Q input , we can write the following relations:
dE ¼ dU þ dK ¼ Q input þ W input
À
Á
dt
ð3:19Þ
Total energy rate d _
E total can be given by
dE
dt
¼
dU
dt
þ
dK
dt
¼
d
dt
ρu þ
1
2
ρe v Á e v
h
i
n
o
dV
ð3:20Þ
where u is the internal strain energy per unit mass. We can write the first law of
thermodynamics remembering that Q input is negative when heat enters the system
from outside and W input is positive when the work is done by the system.
Since
d _
U ¼ Q input þ W input À
dK
dt
ð3:21Þ
d
dt
Z
V
ρudV ¼
Z
A
À q Á ndA þ
Z
V
ρrdV
2
6
4
3
7
5
þ
d
dt
Z
V
1
2
ρe v Á e vdV þ
Z
V
σ : DdV
2
6
4
3
7
5 À
d
dt
Z
V
1
2
ρ e v Á e vdV
ð3:22Þ
Kinetic energy terms cancel each other. Using divergence theorem, we can
transform surface integral to volume integral:
Z
A
q Á ndA ¼
Z
V
∇ Á qdV
ð3:23Þ
We finally collect all terms on the one side and set it to equal to zero:
Z
V
ρ
du
dt
þ ∇ Á q À ρr À σ : D
!
dV ¼ 0
ð3:24Þ
In indicial notation, energy conservation law can be written as
ρ
du
dt
¼ σ ij D ij þ ρr À
∂qj
x j
ð3:25Þ
3.2 First Law of Thermodynamics
79