I
W input dt 6 ¼ 0
ð3:2Þ
and
I
Q input dt 6 ¼ 0
ð3:3Þ
However,
I
W input þ Q input
Â
Ã
dt ¼ 0
ð3:4Þ
Here, exact differential means that there exists a continuously differentiable
function called the potential function. In physical terms, if a system is loaded and
plastically deformed and then returned to its initial state, W input and Q input will not be
recovered completely.
H
dt denotes the integral through the loading cycle. However,
the total energy of the system is an exact differential:
I
du þ dk
ð
Þ dt ¼ 0
ð3:5Þ
3.2.1 Work Done on the System (Power Input)
The work done on the system by external (forces) surface tractions and body forces
can be given by
W input ¼
Z
A
σ
n
ð Þ
Á v dA þ
Z
V
ρb Á v dV
ð3:6Þ
where σ
(n) is the external surface traction per unit area, e v is the velocity field in the
body, b is the per unit mass body forces, and ρ is the unit weight. Concentrated point
loads are also represented by surface tractions. We can write this equation in indicial
notation as
W input ¼
Z
A
σ
n
ð Þ
j e v j dA þ
Z
V
ρ b j e v j dV
ð3:7Þ
The relation between external surface traction σ
n
ð Þ
j and internal stresses σ ij is given
by
76
3 Thermodynamics