Biomedical Engineering Reference
In-Depth Information
Using the above results for the force components it follows that
y
2
2
y
2
2
−
x
+
∂σ
yx
∂
y
h
+
∂σ
yx
∂
y
h
σ
yx
(
x
0
,
y
0
)
h
y
+
σ
yx
(
x
0
,
y
0
)
h
y
2
h
x
y
x
2
2
+
σ
xy
(
x
0
,
y
0
)
h
x
+
∂σ
yx
∂
=
y
2
σ
xy
(
x
0
,
y
0
)
h
x
+
∂σ
xy
∂
h
x
x
y
h
x
2
2
+
∂σ
xy
∂
y
+
∂σ
xy
∂
x
h
x
=
0.
(8.28)
Neglecting terms of order
x
3
,
y
3
etc., reveals immediately that
σ
yx
=
σ
xy
.
(8.29)
Based on this result, the equilibrium equations Eqs. (
8.24
) and (
8.25
) may be
rewritten as
∂σ
xx
∂
x
+
∂σ
xy
∂
y
=
0
(8.30)
∂σ
xy
∂
x
+
∂σ
yy
∂
y
=
0.
(8.31)
Note, that strictly speaking the resulting forces on the faces of the prism as visu-
alized in Fig.
8.7
are not exactly located in the midpoints of the faces. This
should be accounted for in the equilibrium of moment. This would complicate
the derivations considerably, but it would lead to the same conclusions.
In the three-dimensional case a number of additional stress components is
present, see Fig.
8.8
. In total there are six independent stress components:
σ
xx
,
σ
xy
,
σ
xz
,
σ
yy
,
σ
yz
and
σ
zz
. As due to moment equilibrium it can be derived that
σ
yx
=
σ
xy
,
σ
zx
=
σ
xz
,
σ
zy
=
σ
yz
.
(8.32)
σ
zz
σ
yz
σ
xz
σ
xz
σ
yz
σ
yy
σ
xy
e
z
σ
xy
σ
xx
e
y
e
x
Figure 8.8
Stresses in three dimensions.
