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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