Biomedical Engineering Reference
In-Depth Information
u
]
T
,
The deformation gradient
F
may be expressed, using (
2.22
), in terms of [
∇
O
which is a matrix of components
½@
u
i
=@
X
a
,as
T
F
¼
1
þ½r
O
u
:
(2.38)
Since
@
X
a
¼
@
@
X
a
¼
@
x
j
F
T
r
O
¼
r
or
x
j
F
ja
(2.39)
@
@
x
j
@
@
it follows from (
2.38
) and a result obtained in Appendix A, namely that the
transpose of a product of matrices is equal to the product of the transposed matrices
in reverse order, [
AB
]
T
B
T
A
T
, that
¼
T
F
¼
1
þ½r
u
F
:
(2.40)
This result may be used as a recursion formula for
F
. In that role this formula for
F can be substituted into itself once,
T
T
T
F
¼
1
þ½r
u
þ½r
u
½r
u
F
(2.41)
and then again and again,
T
T
T
T
T
T
F
¼
1
þ½r
u
þ½r
u
½r
u
þ½r
u
½r
u
½r
u
þ
h
:
o
:
t
:;
(2.42)
where h.o.t. stands for “higher order terms.” If the terms of second order according
to the criterion (
2.37
) are neglected, the
F
is approximated by
T
F
1
þ½r
u
:
(2.43)
From (
2.22
) it is known that
F
1
T
¼
1
½r
u
;
(2.44)
F
1
a formula that is accurate in the approximation because
F
¼
1
when terms of
second order are neglected.
Two important conclusions may be made from this result. First, for infinitesimal
motions the difference between the use of material and spatial coordinates is
insignificant,
thus
X
and
x
are equivalent as are the gradient operators
∇
O
and
∇
. Concerning these operators note from (
2.39
) that
F
T
r
O
u
¼
½r
u
(2.45)
Search WWH ::
Custom Search