Biomedical Engineering Reference
In-Depth Information
s
x
(
h
l
)
e
x
=
σ
xx
(
h
ln
x
)
e
x
+
σ
xy
(
h
ln
y
)
e
x
,
(8.39)
where
h
denotes the thickness of the prism in the
z
-direction. Dividing by the area
h
l
yields
s
x
=
σ
xx
n
x
+
σ
xy
n
y
.
(8.40)
A similar exercise in the
y
-direction gives
s
y
(
h
l
)
e
y
=
σ
yy
(
h
ln
y
)
e
y
+
σ
xy
(
h
ln
x
)
e
y
,
(8.41)
hence dividing by
h
l
yields
s
y
=
σ
yy
n
y
+
σ
xy
n
x
.
(8.42)
So, the stress vector
σ
xy
via the normal
n
to the infinitesimal surface element at which
s
acts. This can also
be written in a compact form by introducing the so-called
stress tensor
σ
. Let this
tensor be defined according to
s
is directly related to the stress components
σ
xx
,
σ
yy
and
σ
=
σ
xx
e
x
e
x
+
σ
yy
e
y
e
y
+
σ
xy
(
e
x
e
y
+
e
y
e
x
) .
(8.43)
In the two-dimensional case, the stress tensor
σ
is the sum of four dyads. The
components of the stress tensor
σ
σ
can be assembled in the stress matrix
according to
,
σ
xx
σ
xy
σ
=
(8.44)
σ
yx
σ
yy
where
σ
yx
equals
σ
xy
, see Eq. (
8.29
). This stress tensor
σ
has been constructed
such that the stress vector
s
(with components
∼
, acting on an infinitesimal surface
element with outward unit normal
n
(with components
∼
) may be computed via
s
=
σ
·
n
.
(8.45)
This follows immediately from
σ
·
n
=
(
σ
xx
e
x
e
x
+
σ
yy
e
y
e
y
+
σ
xy
(
e
x
e
y
+
e
y
e
x
))
·
n
=
σ
xx
e
x
e
x
·
n
+
σ
yy
e
y
e
y
·
n
+
σ
xy
(
e
x
e
y
·
n
+
e
y
e
x
·
n
)
=
σ
xx
n
x
e
x
+
σ
yy
n
y
e
y
+
σ
xy
(
n
y
e
x
+
n
x
e
y
)
=
(
σ
xx
n
x
+
σ
xy
n
y
)
e
x
+
(
σ
yy
n
y
+
σ
xy
n
x
)
e
y
.
(8.46)
Hence, it follows immediately that, with
s
=
s
x
e
x
+
s
y
e
y
:
s
x
=
σ
xx
n
x
+
σ
xy
n
y
(8.47)
and
s
y
=
σ
xy
n
x
+
σ
yy
n
y
.
(8.48)
