Biomedical Engineering Reference
In-Depth Information
@
2
E
mq
x
k
þ
@
2
E
ik
x
m
@
2
E
kq
x
m
@
2
E
im
x
0
q
Þ
0
¼
ð
x
q
(2.68)
@
x
i
@
@
x
q
@
@
x
i
@
@
x
q
@
x
k
is satisfied only when the compatibility conditions (
2.54
), or equivalently (
2.54
)or
(
2.55
)or
0 is a necessary and sufficient
condition that the integration of the strain-displacement relations will yield a
single-valued and continuous displacement field.
r
E
r¼
0, hold. Thus
r
E
r¼
Problems
2.4.1. For the motions of the form (
2.10
) given in Problem 2.1.1, namely 2.1.1(a)
through 2.1.1(g), determine if the infinitesimal strain tensors,
E
, calculated in
2.3.1 satisfy the conditions of compatibility.
2.4.2. Is the following strain state possible for an object in which the displacement
field must be continuous and single valued? Justify your answer analytically.
2
3
x
3
ð
x
1
þ
x
2
Þ
x
1
x
2
x
3
0
4
5
:
e
¼
c
x
1
x
2
x
3
x
3
x
2
0
0
0
0
2.4.3. Demonstrate the validity of the formula (
2.64
) by substituting the formulas
relating
E
and
Y
to the displacement gradients (
2.49
) into (
2.64
) and show
that an identity is obtained. This is more easily done in the indicial notation.
2.4.4. Verify that substitution of the formula (
2.67
) into (
2.66
) leads to the result
(
2.68
). This is much more easily done in the indicial notation.
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