Biomedical Engineering Reference
In-Depth Information
direction. The next step in this development is to cancel A throughout (
3.18
) and
allow the plane whose normal is
n
to approach
P
, causing the volume of the
tetrahedron to vanish as
h
tends to zero. Before doing this, note that since
Q
must
always lie on
A
,as
h
tends to zero, by the mean value theorem:
t
ðnÞ
ð
P
Þ¼
limit
h!
0
t
ðnÞ
1
ð
P
Þ¼
limit
h!
0
t
ðnÞ
1
ð
Q
Þ;
T
ð
P
Þ¼
limit
h!
0
T
ð
P
Þ¼
limit
h!
0
T
ð
Q
Þ:
(3.19)
Canceling
A
throughout (
3.18
) and taking the limiting process as
h
tends to zero,
noting that the object force and the acceleration vanish as the volume of the
tetrahedron vanishes, it follows from (
3.18
) and (
3.19
) that
t
ðnÞ
1
¼
T
11
n
1
þ
T
12
n
2
þ
T
13
n
3
:
(3.20)
Repeating this analysis for the
e
2
and
e
3
directions the result (
3.15
)is
established.
Thus we have shown that the double infinity of possible stress vectors
t
(
n
)
, which
constitutes a state of stress at a point in an object, can be completely characterized by
the nine components of
T
. These nine components are simply the three components
of three different stress vectors, one acting on each of the coordinate planes of a
reference frame. Thus, in the matrix of tensor components (
3.13
) the first row consists
of the components of the stress vector acting on a plane whose normal is in the
e
1
direction. A similar interpretation applies to the second and third rows. When the
meaning is not obscured, we will drop the subscript (
n
) in the equation
t
(
n
)
¼
T
n
and
write it as
t
n
, with it being understood that the particular
t
depends upon
n
. The
normal stress on a plane is then given by
t
¼
T
n
¼
n
T
n
and the shear stress in a
direction
m
lying in the plane whose normal is
n
,
m
n
¼
0, is given by
t
m
¼
m
T
n
m
. Note that if a vector
m
0
is introduced that reverses the direction of
m
,
m
0
¼
¼
n
T
m
0
¼
m
0
m
0
¼
m
then the associated shear stress is given by
t
T
n
¼
n
T
m
. This shows that, if the unit vector
m
is reversed in direction, the opposite value
of the shear stress is obtained. Note also that if the unit vector
n
is reversed in
direction, the opposite value of the shear stress is obtained. If both the unit vectors
n
and
m
are reversed in direction, the sign of the shear stress is unchanged. These
conclusions are all consistent with the definition of the sign of the shear stress.
A short calculation will show that the stress matrix
T
is a tensor. Recall that, in
order for
T
to be a tensor, its components in one coordinate system had to be related
to the components in another coordinate system by
t
T
ðLÞ
¼
T
ðGÞ
Q
T
and
T
ðGÞ
¼
Q
T
T
ðLÞ
Q
Q
:ð
A83
Þ
repeated
To show that the
T
in the relationship (
3.15
),
t
¼
T
n
, has the tensor property,
n
is specified in the Latin coordinate system,
t
(
L
)
T
(
L
)
n
(
L
)
.
the equation
t
¼
T
¼
Then, using the vector transformation law (A77) for
t
and
n
,
t
(
L
)
t
(
G
)
and
¼
Q
n
(
L
)
n
(
G
)
, respectively, the expression
t
(
L
)
T
(
L
)
n
(
L
)
¼
Q
¼
is then rewritten as
t
(
G
)
T
(
L
)
n
(
G
)
,or
t
(
G
)
Q
T
T
(
L
)
n
(
G
)
. Finally,
Q
¼
Q
¼
Q
it can be noted from
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