Biomedical Engineering Reference
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stress vector,
t
A
ðnÞ
(
P
), as
D
A
(i)
tends to zero through a sequence of progressively
smaller areas,
D
A
(1)
,
D
A
(2)
,
D
A
(3)
,
...
,
D
A
(N)
,
...
, all containing the point
P
,
t
A
t
ðnÞ
ð
P
Þ¼
limit
D
ðnÞ
ð
P
Þ:
(3.11)
A
ð
i
Þ
!
0
When a similar limit is applied to the average couple stress vector we assume that
the limit is zero:
c
A
0
¼
limit
D
ðnÞ
ð
P
Þ:
(3.12)
A
ð
i
Þ
!
0
In effect we are assuming that the forces involved are of finite magnitude and
that the moment arm associated with
A
(i)
tends to zero. For
almost all continuum theories the assumption (
3.12
) is adequate.
The internal force interaction at a point in an object is adequately represented by
the stress vector
t
(
n
)
across the plane whose normal is
n
. However, there is a double
infinity of distinct planes with normals
n
passing through a single point; thus there
is a double infinity of distinct stress vectors acting at each point. The multitude of
stress vectors,
t
(
n
)
, at a point is called the state of stress at the point. The totality of
vectors
t
(
n
)
(
P
) at a fixed point
P
, and for all directions
n
, is called the
state of stress
at the point
P
. The representation of the state of stress at a point is simplified by
proving that
t
(
n
)
(
P
) must be a linear function of the vector
n
, as will be done below.
The coefficients of this linear relationship will be the stress tensor
T
. Thus
T
will be
a linear transformation that transforms
n
into
t
(
n
)
,
t
(
n
)
(
P
)
D
m
(i)
vanishes as
D
n
. The proof that
T
is, in fact, a tensor and the coefficient of a linear transformation will also be
provided below. However, even though it has not yet been proved,
T
will be
referred to as a tensor. The stress tensor
T
has components relative to an orthonor-
mal basis that are the elements of the matrix
¼
T
(
P
)
2
4
3
5
:
T
11
T
12
T
13
T
21
T
22
T
23
T
31
T
32
T
33
T
¼
(3.13)
The component
T
ij
of the stress tensor is the component of the stress vector
t
(
n
)
¼
t
(j)
acting on the plane whose normal
n
is in the
e
j
direction,
n
¼
e
j
, projected
in the
e
i
direction,
T
ij
¼
e
i
t
ð
j
Þ
:
(3.14)
The nine components of the stress tensor defined by (
3.14
) and represented by
(
3.13
) are therefore the
e
1
,
e
2
, and
e
3
components of the three stress vectors
t
(i)
,
t
(j)
,
and
t
(k)
which act at the point
P
, on planes parallel to the three mutually perpendic-
ular coordinate planes. This is illustrated in Fig.
3.3
. The components
T
11
,
T
22
,
T
33
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