Biomedical Engineering Reference
In-Depth Information
configuration and d
v
is an element of volume in the final configuration. Substituting
the relationship (
11.36
) between d
v
and d
V
,d
v
¼
J
d
V
, into (
11.43
) it follows that
ð
O
r
R
ð
ð
O
rð
M
¼
X
Þ
dV
¼
x
;
t
Þ
JdV
:
(11.44)
Since this same result must hold for each and every part of the object we
conclude that the relationship between the two density fields is given by the simple
formula
r
R
¼
J
r;
(11.45)
which is an alternative statement of mass balance. The argument that is used to go
from (
11.44
)to(
11.45
) is the same argument that was used to go from (3.4) to (3.5),
referred to the two different configurations, a similar procedure to the arguments
leading to the result
above is followed. The total force
f
acting on an
object, or on any particular subpart of the object, is considered. The total force is the
same in both configurations; and therefore the product of the stress and a differen-
tial area element integrated over the object must be the same in both configurations.
Thus
r
R
¼
J
r
ð
ð
T
1
PK
f
¼
dA
¼
T
da
;
(11.46)
@O
@O
where
T
1
PK
and d
A
are the stress tensor and differential area element in the initial
configuration and
T
and d
a
are the stress and differential area in the instantaneous
or deformed configuration.
T
is the Cauchy stress, of course. When the relationship
of Nanson between d
a
and d
A
given by (
11.42
) is substituted into (
11.46
) we find
that
ð
ð
T
T
1
PK
F
1
f
¼
dA
¼
JT
ð
Þ
dA
:
(11.47)
@
O
@
O
Since this must hold for all parts of the object, the same argument as in the
transitions from (
11.44
)to(
11.45
) and (3.4) to (3.5), it may be concluded that
T
T
1
PK
JT ðF
1
J
1
T
1
PK
F
T
¼
Þ
; e
or
T ¼
:
(11.48)
T
1PK
is called the
first Piola-Kirchhoff or
Lagrangian stress tensor. The relation
of Cauchy involving the Cauchy stress tensor and the spatial reference frame,
namely that the stress vector
t
(
n
)
acting on any plane whose normal is
n
is given
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