According to the conservation of momentum principle, the time rate of change of
the total momentum equals the vector sum of all the external forces:
Z
A
σ
n
ð Þ dA þ
Z
V
ρb dV ¼
d
dt
Z
ρe v dV
Using Cartesian coordinate system where x, y, and z are represented by indices i
yields
Z
A
σ
n
ð Þ
i dA þ
Z
V
ρb i dV ¼
d
dt
Z
ρ e
v i dV
Components of external surface loading vector σ
(n) can be given by
σ
n
ð Þ
i ¼ σ ij n j
or in Cartesian coordinates
σ
n
ð Þ
x ¼ σ xx n x þ σ xy n y þ σ xz n z
σ
n
ð Þ
y ¼ σ yx n x þ σ yy n y þ σ yz n z
σ
n
ð Þ
z ¼ σ zx n x þ σ zy n z þ σ zz n z
where n x, n x, n x, are direction cosines between the normal to the arbitrary plane and
spatial coordinates. In matrix notation, this can be written as
σ
n
ð Þ
n
o
¼ n
f g σ
½ Š
σ
n
ð Þ
È
É ¼ σ
n
ð Þ
x σ
n
ð Þ
y σ
n
ð Þ
z
n
o
, {n} ¼ {n x n y n z }, σ
½ Š ¼
σ xx σ xy σ xz
σ yx σ yy σ yz
σ zx σ zy σ zz
2
6
4
3
7
5
[σ] is a second-order Cauchy stress tensor, which is a linear vector function.
Using the divergence theorem, we can write the following conversion:
Z
A
σ
n
ð Þ
i dA ¼
Z
V
∂σ ji
∂x j
dV
Meanwhile, due to conservation of mass, material time derivative of volume
integral can be written as (Malvern 1969)
2.17 Conservation of Momentum Principle
67