Biomedical Engineering Reference
In-Depth Information
r
E
0
(2.53)
or in the index notation as
2 E jm
e ijk e pmn @
x n ¼
0
(2.54)
@
x k @
or in scalar form as the following six equations:
@
2 E 11
x 3 ¼ @
x 1 @
E 23
@
x 1 þ @
E 31
@
x 2 þ @
E 12
@
2 @
2 E 12
x 2 ¼ @
2 E 11
@
x 2 þ @
2 E 22
@
;
;
@
x 2 @
@
x 3
@
x 1 @
x 1
2 E 22
2 E 23
2 E 22
@
2 E 33
@
@
x 1 ¼ @
x 2 @
E 31
@
x 2 þ @
E 12
@
x 3 þ @
E 23
@
2 @
x 3 ¼ @
x 3 þ @
;
;
(2.55)
x 2
@
x 3 @
@
x 1
@
x 2 @
2 E 33
2 E 31
2 E 33
@
2 E 11
@
@
x 2 ¼ @
x 3 @
E 12
@
x 3 þ @
E 23
@
x 1 þ @
E 31
@
2 @
x 1 ¼ @
x 1 þ @
;
:
x 3
@
x 1 @
@
x 2
@
x 3 @
Equations ( 2.53 ) and ( 2.54 ) are symmetric second rank tensors in three
dimensions and therefore have the six components given by ( 2.55 ). It follows that
each of the six scalar equations ( 2.55 ) must be satisfied in order to insure compati-
bility. The conditions ( 2.53 ) are a direct consequence of the definition of strain, that
is to say that E
u ) T
¼
(1/2)((
þ ∇
u )
¼
(1/2)( u
∇ þ ∇
u ) implies
that
r
E
0. To see that this is true, consider the result of operating on
E
¼
(1/2)( u
∇ þ ∇
u ) from the left by
r
and from the right by
r
; one
obtains the expression
2
ðr
E
rÞ¼r
u
rrþrr
u
r:
(2.56)
, which occurs in both terms on the right hand side of ( 2.56 )
is called the “curl grad”; the curl of the gradient applied to a function f is zero,
rr
The operator
rr
f
¼
0. In the indicial notation this is easy to see,
rr
f
¼
e ijk ð@
f
=@
x j @
x k Þ
e i ¼
0, because of the symmetry of the indices on the partial derivatives and skew-
symmetry in the components of the alternator (see Appendix A.8). Both terms on the
right hand side of ( 2.56 ) contain the operator curl grad,
rr
, applied to a function,
hence
r
E
0. It may also be shown that the reverse is true, namely that
u ) T
r
E
0 implies that E
¼
(1/2)((
þ ∇
u ). Thus E
¼
(1/2)
u ) T
((
0.
In order to both prove and motivate this result consider the two integration paths
from the point P o to the point P 0 in an object (Fig. 2.11 ). 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 (see Appendix A.15 Exact differentials). Recall the theorem at the
þ ∇
u ) is a necessary and sufficient condition that
r
E
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