CONTINUUM MECHANICS for ENGINEERS

is known as the stress work, and its integrand σ as the stress power. The
ijDij balance of mechanical energy given by Eq 5.7-4 shows that, of the total work
done by the external forces, a portion goes toward increasing the kinetic
energy, and the remainder appears as work done by the internal stresses.
In general, S cannot be expressed as the material derivative of a volume
integral, that is,
d
S ≠ () dV
(5.7-6)
dt V
because there is no known function we could insert as the integrand of this
equation. However, in the special situation when
S = U= ˙ ˙
(5.7-7)
d
ρudV = ρudV
dt V V
where U is called the internal energy and u the specific internal energy, or energy
density (per unit mass), Eq 5.7-4 becomes
or, briefly,
d
dt
∫
∫ ∫
∫ ( 2 i i )
i i
i i
V ∫V∫ S
1 ρ v v + udV = ρbvdV
= t
( ˆn )
v dS
K˙ + U˙
(5.7-8a)
= P (5.7-8b)
(The symbol u is used for specific internal energy because of its widespread
acceptance in the literature. There appears to be very little chance that it
might be misinterpreted in this context as the magnitude of the displacement
vector u). We note that Eq 5.7-8 indicates that part of the external work P
causes an increase in kinetic energy, and the remainder is stored as internal
energy. As we shall see in Chapter Six, ideal elastic materials respond to
forces in this fashion.
For a thermomechanical continuum, we represent the rate at which thermal
energy is added to a body by
Q = ρrdV − qinidS (5.7-9)
∫V
∫S
The scalar field r specifies the rate at which heat per unit mass is produced
by internal sources and is known as the heat supply. The vector q i, called the
heat flux vector, is a measure of the rate at which heat is conducted into the
body per unit area per unit time across the element of surface dS whose