ΔV ¼
1
3
hΔA
Substituting h in the previous equation yield
ΔV ¼
1
3
OA Á cos α
ð
Þ ΔA ¼
1
3
OB Á cos β
ð
Þ ΔA ¼
1
3
OC Á cos γ
ð
Þ ΔA
Note that we can define the direction cosines also using the relationship between
the areas of the faces of the tetrahedron:
cos α ¼
ΔA X
ΔA
because ΔA X is the projection of the inclined area of interest ΔA on Y À Z plane
with X axis as its normal. Hence,
ΔV ¼
1
3
OA ΔA X
Similarly, we can write
ΔV ¼
1
3
OB ΔA Y and ΔV ¼
1
3
OC ΔA Z
Substituting ΔV, ΔA X , ΔA Y , and ΔA Z in conservation of momentum equation, we
obtain
σ
n
ð Þ
ΔA
ð Þþρb
1
3
hΔA
¼ σ
X
ð Þ
ΔA n X þ σ
Y
ð Þ
ΔA n Y þ σ
Z
ð Þ
ΔA n Z þ ρ
1
3
hΔA
dv
dt
Eliminating ΔA from each term leads to
σ
n
ð Þ
þ ρb
1
3
h ¼ σ
X
ð Þ n X þ σ
Y
ð Þ n Y þ σ
Z
ð Þ n Z þ ρ
1
3
h
dv
dt
Let the height of the tetrahedron h go to zero to define our imaginary point. The
terms with h disappear:
σ
n
ð Þ
¼ σ
X
ð Þ n X þ σ
Y
ð Þ n Y þ σ
Z
ð Þ n Z
ð3:1Þ
This equation defines the stress at a point on an arbitrary oblique plane passing
through point Q. This equation was derived from the conservation of momentum
principle of a collection of particles. Hence, it applies to solid mechanics as well as
fluid mechanics. However, it important to point out that this is not an equilibrium of
stress equation. There is an equilibrium of forces but not stresses. If direction cosines
2.2 Stress
17