Biomedical Engineering Reference
In-Depth Information
The final step in this development is to employ the same argument as was
employed in the transition from (
3.4
)to(
3.5
); see the discussion following (
3.5
).
It follows then that the stress tensor is symmetric,
T
T
T
¼
;
(3.37)
at each point in the object
O
.
The final form of the stress equations of motion (Truesdell and Toupin,
1960
)is
obtained from the combination of (
3.37
) and (
3.30
), thus
T
T
r
€
x
¼r
T
þ r
d
;
T
¼
:
(3.38)
This local statement of Newton's second law retains aspects of the original. The
mass times acceleration is represented by density times acceleration on the left-
hand side. The sum of the forces is represented on the right-hand side by the
gradient of the stress tensor and the action-at-a-distance force. The expanded scalar
version of the stress equations of motion is
x
1
¼
@
T
11
@
x
1
þ
@
T
12
@
x
2
þ
@
T
13
@
r€
x
3
þ r
d
1
;
x
2
¼
@
T
12
@
x
1
þ
@
T
22
@
x
2
þ
@
T
23
@
r€
x
3
þ r
d
2
;
(3.39)
x
3
¼
@
T
13
@
x
1
þ
@
T
23
@
x
2
þ
@
T
33
@
r€
x
3
þ r
d
3
;
where the symmetry of the stress tensor is expressed in the subscripted indices. For
a two-dimensional motion the stress equations of motion reduce to
x
1
¼
@
T
11
@
x
1
þ
@
T
12
@
x
2
¼
@
T
12
@
x
1
þ
@
T
22
@
r€
x
2
þ r
d
1
; r€
x
2
þ r
d
2
:
(3.40)
Example 3.4.1
The stress tensor in an object is given by
2
4
3
5
;
c
1
x
1
þ
c
2
x
2
c
4
x
1
c
1
x
2
0
T
¼
c
4
x
1
c
1
x
2
c
3
x
1
þ
c
4
x
2
þ r
gx
2
0
0
0
c
5
where
c
i
,
i
,5, are constants. This same object is subjected to an action-at a-
distance force
d
with components [0,
¼
1,
...
g, 0]. Determine the components of the
acceleration vector of this object.
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