2.7 Conservation of momentum
The law of conservation of momentum can be
written

(2.19) 
where
represents any body force
per unit mass. Body forces represent any force
which does not act at a bounding surface, including those that act
at a distance, such as gravitational force.
The equation is derived by considering the time
rate of change of the momentum of a mass of particles. We consider
a volume of material of ﬁxed
mass moving through space and therefore present rate of
change by the material derivative .
Applying Gauss’s theorem
and Eq. (2.16
), the surface force is

(2.20) 
Equating the rate of change of momentum to the forces and, noting
mass is ﬁxed so
is constant in time, gives

(2.21) 
The integrand must equal 0, resulting in Eq. (
2.19).
Divergence was described in section 2.4
as the ﬂux across a surface per
unit volume as . The divergence of stress
similarly represents the stress ﬂux across the surface,
i.e. force, per unit volume
as ,
given by

(2.22) 
Eq. (
2.19
) speciﬁcally relates to
linear momentum. Instead,
conservation of
angular
momentum
, in the absence of any “couple stresses” that
generate a moment ﬁeld, is given by

(2.23) 
The derivation of Eq. (
2.23
) is fairly complex so is
omitted here.
Tensor symmetry
Symmetry in a tensor refers to components
being symmetric about the diagonal, i.e. , and . The transpose of a tensor, denoted by the
‘’
superscript, switches components across the diagonal such that:

(2.24) 
is
therefore
symmetric if
. A
skew (antisymmetric) tensor has
. A tensor can be
decomposed into symmetric and skew parts by:

(2.25) 