Biomedical Engineering Reference
In-Depth Information
In this case the normal on a plane has three components:
n
=
n
x
e
x
+
n
y
e
y
+
n
z
e
z
,
therefore
s
=
σ
·
n
=
σ
xx
e
x
e
x
+
σ
yy
e
y
e
y
+
σ
zz
e
z
e
z
+
σ
xy
(
e
x
e
y
+
e
y
e
x
)
+
σ
xz
(
e
x
e
z
+
e
z
e
x
)
e
y
)
·
+
σ
yz
(
e
y
e
z
+
e
z
(
n
x
e
x
+
n
y
e
y
+
n
z
e
z
)
=
σ
xx
n
x
e
x
+
σ
yy
n
y
e
y
+
σ
zz
n
z
e
z
+
σ
xy
(
n
y
e
x
+
n
x
e
y
)
+
σ
xz
(
n
z
e
x
+
n
x
e
z
)
+
σ
yz
(
n
z
e
y
+
n
y
e
z
) .
(8.56)
Hence the components of the stress vector
s
=
s
x
e
x
+
s
y
e
y
+
s
z
e
z
satisfy
s
x
=
σ
xx
n
x
+
σ
xy
n
y
+
σ
xz
n
z
s
y
=
σ
xy
n
x
+
σ
yy
n
y
+
σ
yz
n
z
s
z
=
σ
xz
n
x
+
σ
yz
n
y
+
σ
zz
n
z
.
(8.57)
Application of Eq. (
8.55
) and using
⎡
⎤
⎡
⎤
s
x
s
y
s
z
n
x
n
y
n
z
⎣
⎦
⎣
⎦
∼
=
,
∼
=
,
(8.58)
gives an equivalent, but much shorter, expression for Eq. (
8.57
):
∼
=
σ
∼
.
(8.59)
If all the shear stress components are zero, i.e.:
σ
xy
=
σ
xz
=
σ
yz
=
0, and all the
normal stresses are equal, i.e.
σ
xx
=
σ
yy
=
σ
zz
, this normal stress is called the
pressure
p
such that
p
=−
σ
xx
=−
σ
yy
=−
σ
zz
,
(8.60)
while
σ
=−
p
(
e
x
e
x
+
e
y
e
y
+
e
z
e
z
)
=−
p
I
,
(8.61)
with
I
the unit tensor. This is illustrated in Fig.
8.13
.
8.5
Principal stresses and principal stress directions
Assume, that in a certain point of the material volume the stress state is known by
specification of the tensor
σ
. One might ask, whether it is possible to chose the
