Biomedical Engineering Reference
In-Depth Information
s
n
s
σ
xx
σ
xy
s
t
σ
xy
σ
yy
Figure 8.11
Stress vector
s
acting on an inclined plane with normal
n
, decomposed into a stress vector normal
and stress vector tangent to the plane.
Eq. (
8.45
) can also be written in column notation as
∼
=
σ
∼
.
(8.49)
, defined as the sum of four dyads,
is to compute the stress vector that acts on an infinitesimally small area that is
oriented in space as defined by the normal
The purpose of introducing the stress tensor
σ
n
. For any given normal
n
this stress
vector is computed via
n
.
At any point in a body and for any plane in that point this stress vector can be
computed. This stress vector itself may be decomposed into a stress vector normal
to the plane (normal stress) and a vector tangent to the plane (shear stress), see
Fig.
8.11
. Hence let
s
=
σ
·
s
=
σ
·
n
,
(8.50)
then the stress vector normal to the plane,
s
n
follows from
s
n
=
n
=
((
σ
·
n
)
·
n
)
n
=
σ
·
n
·
(
nn
) .
(
s
·
n
)
(8.51)
The stress vector tangent to the plane is easily obtained via
s
=
s
n
+
s
t
,
(8.52)
hence
s
t
=
s
−
s
n
=
σ
·
n
−
(
(
σ
·
n
)
·
n
)
n
=
σ
·
n
·
(
I
−
nn
) .
(8.53)
Example 8.3
If the stress state is specified as depicted in Fig.
8.12
, then
σ
=
10
e
x
e
x
+
3(
e
x
e
y
+
e
y
e
x
).
