Biomedical Engineering Reference
In-Depth Information
This means that
σ
xx
may only be a function of
y
:
σ
xx
=
σ
xx
(
y
). Likewise,
σ
yy
may
only be a function of
x
.
8.4
Stress tensor
Suppose that, in a two-dimensional configuration, all three stress components (
σ
xx
,
σ
xy
and
σ
yy
) are known. How can the resulting stress vector acting on an arbitrary
cross section of a body be computed based on these stress components? To answer
this question, consider an arbitrary, but infinitesimally small prism, having a trian-
gular cross section as depicted in Fig.
8.10
(a). Two faces of the prism are parallel
to
e
x
and
e
y
, respectively, while the third face, having length
l
, is oriented at
some angle
α
with respect to
e
x
. The orientation in space of this face is fully
characterized by the unit outward normal vector
n
, related to
α
by
n
=
n
x
e
x
+
n
y
e
y
=
sin(
α
)
e
x
+
cos(
α
)
e
y
.
(8.36)
The face parallel to
e
x
has length
h
x
=
n
y
l
, while the face parallel to
e
y
has
length
h
y
=
n
x
l
. This follows immediately from
h
y
l
→
n
x
=
sin(
α
)
=
h
y
=
n
x
l
,
(8.37)
while
h
x
n
y
=
cos(
α
)
=
l
→
h
x
=
n
y
l
.
(8.38)
The stresses acting on the left and bottom face of the triangular prism are
depicted in Fig.
8.10
(b). On the inclined face a stress vector
s
is introduced. Force
equilibrium in the
x
-direction yields
s
y
e
y
s
n
y
e
y
n
σ
xx
s
x
e
x
Δ
l
n
x
e
x
σ
xy
h
y
=
n
x
Δ
l
σ
xy
α
σ
yy
h
x
=
n
y
Δ
l
(a)
(b)
Figure 8.10
Stress vector
s
acting on an inclined plane with normal
n
.
