Biomedical Engineering Reference
In-Depth Information
The equilibrium equations Eqs. (
8.30
and (
8.31
) can be generalized to three
dimensions as
∂σ
xx
∂
x
+
∂σ
xy
y
+
∂σ
xz
=
0
(8.33)
∂
∂
z
∂σ
xy
∂
x
+
∂σ
yy
y
+
∂σ
yz
=
0
(8.34)
∂
∂
z
+
∂σ
yz
∂
∂σ
xz
∂
+
∂σ
zz
∂
=
0.
(8.35)
x
y
z
Some interpretation of the equilibrium equations is given by considering a number
of special cases.
Example 8.1
If all the individual partial derivatives appearing in Eqs. (
8.30
) and (
8.31
) are zero,
that is if
∂σ
xx
∂
x
=
∂σ
xy
∂
y
=
∂σ
yy
∂
y
=
∂σ
xy
∂
x
=
0,
0,
0,
0,
the stresses on the faces of the cube are shown in Fig.
8.9
. Clearly the forces on
opposing faces are in equilibrium, as demanded by the equilibrium equations if
the individual partial derivatives are zero.
Example 8.2
In the absence of shear stresses (
σ
xy
=
0), or if the shear stresses are constant
(
σ
xy
=
c
) it follows that
∂σ
xy
∂
x
=
∂σ
xy
∂
y
=
0.
Hence the equilibrium equations reduce to:
∂σ
xx
∂
∂σ
yy
∂
x
=
0,
y
=
0.
σ
yy
σ
xy
σ
xx
σ
xx
σ
xy
σ
xy
σ
xy
σ
yy
Figure 8.9
Stresses on the faces.
