Biomedical Engineering Reference
In-Depth Information
8
Stress in three-dimensional
continuous media
In this chapter the concepts introduced in Chapter
6
for a one-dimensional con-
tinuous system are generalized to two-dimensional configurations. Extension to
three-dimensional problems is briefly discussed. First the equilibrium conditions
in a two- or three-dimensional body are derived from force equilibrium of an
infinitesimally small volume element. Thereafter, the concept of a stress tensor, as
a sum of dyads, is introduced to compute the stress vector acting on an arbitrary
surface in a material point of the body.
8.1
Stress vector
Before examining the equilibrium conditions in a two-dimensional body, the
concept of a stress vector is introduced. For this purpose we consider an
infinitesimally small surface element having area
A
, see Fig.
8.1
.
On this surface an infinitesimally small force vector
F
is applied with com-
ponents in the
x
-,
y
- and
z
-direction:
F
=
F
x
e
x
+
F
y
e
y
+
F
z
e
z
. Following
the definition of stress, Eq. (
6.8
), three stresses may be defined:
A
→
0
F
x
s
x
=
lim
A
,
(8.1)
that acts in the
x
-direction, and
A
→
0
F
y
s
y
=
lim
A
,
(8.2)
that acts in the
y
-direction, and
A
→
0
F
z
s
z
=
lim
A
,
(8.3)
that acts in the
z
-direction. Hence, a
stress vector
may be defined:
s
=
s
x
e
x
+
s
y
e
y
+
s
z
e
z
.
(8.4)
