Δm
dv
dt
¼ ρΔV
ð
Þ
dv
dt
It is assumed that Δm does not change by time. The conservation of momentum
principle yields the following equilibrium equation for the free body (Fig. 2.6):
σ
n
ð Þ
ΔA þ ρbΔV À σ
X
ð Þ
ΔA X À σ
Y
ð Þ
ΔA Y À σ
Z
ð Þ
ΔA Z ¼ ρΔV
ð
Þ
dv
dt
We need to calculate the height and volume of the arbitrary tetrahedron at point
Q. Then we let the height h go to zero, to be able define an imaginary point Q:
cosα ¼
h
OA
, cosβ ¼
h
OB
, cosγ ¼
h
OC
n X ¼ cosα, n Y ¼ cosβ, n Z ¼ cosγ
Three direction cosines, n X , n Y , n z , are also defined by
n
2
X þ n
2
y þ n
2
z ¼ 1
As a result,
h ¼ OA Á n X ¼ OB Á n Y ¼ OC Á n Z
The volume of the pyramid is given by
Y
X
Z
A
B
C
O
n
α
β
γ
N
h
γ
i
j
Y
X
β
α
Z
k
Fig. 2.6 Geometry of the tetrahedron
16
2 Stress and Strain in Continuum
dv
dt
¼ ρΔV
ð
Þ
dv
dt
It is assumed that Δm does not change by time. The conservation of momentum
principle yields the following equilibrium equation for the free body (Fig. 2.6):
σ
n
ð Þ
ΔA þ ρbΔV À σ
X
ð Þ
ΔA X À σ
Y
ð Þ
ΔA Y À σ
Z
ð Þ
ΔA Z ¼ ρΔV
ð
Þ
dv
dt
We need to calculate the height and volume of the arbitrary tetrahedron at point
Q. Then we let the height h go to zero, to be able define an imaginary point Q:
cosα ¼
h
OA
, cosβ ¼
h
OB
, cosγ ¼
h
OC
n X ¼ cosα, n Y ¼ cosβ, n Z ¼ cosγ
Three direction cosines, n X , n Y , n z , are also defined by
n
2
X þ n
2
y þ n
2
z ¼ 1
As a result,
h ¼ OA Á n X ¼ OB Á n Y ¼ OC Á n Z
The volume of the pyramid is given by
Y
X
Z
A
B
C
O
n
α
β
γ
N
h
γ
i
j
Y
X
β
α
Z
k
Fig. 2.6 Geometry of the tetrahedron
16
2 Stress and Strain in Continuum