CONTINUUM MECHANICS for ENGINEERS

of a body with respect to a given point is equal to the moment of the surface
and body forces with respect to that point. For the body shown in Figure 5.1,
if we take the origin as the point of reference, the angular momentum
principle has the mathematical form
d
dt
(5.6-1)
Making use of Eq 5.3-11 in taking the derivative on the left-hand side of the
equation and applying the divergence theorem to the surface integral after
introducing the identity t
( ˆn )
= σ n results in
which reduces to
∫ ijk j k
ijk j k
V ∫S∫V ε x ρvdV = ε x t
( ˆn )
dS + ε x ρbdV
∫
V
k qk q
[ ]
εijk xj( ρv˙k−σqk,q −ρbk)−σ
jk dV
∫
ε σ dV
V
ijk kj
ijk j k
= 0
= 0 (5.6-2)
because of Eq 5.4-4 (the equations of motion) and the sign-change property
of the permutation symbol. Again, with V arbitrary, the integrand must
vanish so that
εijkσkj = 0 (5.6-3)
which by direct expansion demonstrates that σ , and the stress tensor
kj =
σ jk
is symmetric. Note that in formulating the angular momentum principle by
Eq 5.6-1 we have assumed that no body or surface couples act on the body.
If any such concentrated moments do act, the material is said to be a polar
material, and the symmetry property of no longer holds. But as mentioned
in Chapter Three, this is a rather specialized situation and we shall not
consider it here.
5.7 Law of Conservation of Energy, The Energy Equation
The statement we adopt for the law of conservation of energy is the following:
the material time derivative of the kinetic plus internal energies is equal
to the sum of the rate of work of the surface and body forces, plus all other
energies that enter or leave the body per unit time. Other energies may
include, for example, thermal, electrical, magnetic, or chemical energies. In