Biomedical Engineering Reference
In-Depth Information
Fig. 2.11
Illustration of two
integration paths from the
point
P
o
to the point
P
0
in an
object. If the result of the
integration from the point
P
o
to the point
P
0
is to be the
same along all paths chosen
between these two points,
then the value of the integral
around any closed path in the
object must be zero. This
means that the integrand of
the integral must be an exact
differential
start of most texts on ordinary differential equations concerning exact differentials:
If
M
(
x
,
y
) and
N
(
x
,
y
) are continuous functions and have continuous partial
derivatives in a region of the
x-y
plane, then the expression
M
(
x
,
y
)
dx
þ
N
(
x
,
y
)
dy
is an exact differential if and only if
x
throughout the region. This
theorem will be applied to prove that the compatibility relations
@
M
=@
y
¼ @
N
=@
r
E
r¼
0 are
both necessary and sufficient conditions for the continuous and single-valued nature
of the displacement field obtained by integration from the strain-displacement
relations. If the displacement vector is known at the point
P
o
then integration of
d
u
from the point
P
o
to the point
P
0
(Fig.
2.11
) will determine
u
(
x
0
), thus,
P
0
ð
P
0
P
o
ðr
ð
x
0
Þ¼
u
o
u
o
T
u
ð
þ
d
u
¼
þ
u
Þ
d
x
:
(2.57)
P
o
Recall from (
2.43
) and (
2.51
) that
T
ðr
u
Þ
¼
E
þ
Y
(2.58)
it follows that
P
0
ð
P
0
ð
x
0
Þ¼
u
o
u
ð
þ
E
d
x
þ
Y
d
x
:
(2.59)
P
o
P
o
The last integral in the previous result may be rewritten as
P
0
ð
P
0
ð
x
0
Þ
Y
d
x
¼
Y
d
ð
x
(2.60)
P
o
P
o
Search WWH ::
Custom Search