Biomedical Engineering Reference
In-Depth Information
the two strain tensors coincide only if terms of the order
t
2
and higher are neglected.
Thus the motion (6) is one of small strain only when all terms of order
t
2
and higher
are neglected.
There are several other strain measures used in the development and analysis of
finite deformations. Two of the most widely used are the right and left
Cauchy-Green tensors,
C
and
B
, respectively. These two tensors are simply the
squares of the right and left stretch tensors,
U
and
V
,
U
2
F
T
V
2
F
T
C
¼
¼
F
;
B
¼
¼
F
:
(11.27)
It is also convenient to introduce the inverse of the left Cauchy-Green tensor
denoted by
c
,
1
B
1
V
2
F
T
c
¼
¼
¼ð
F
Þ
:
(11.28)
The Lagrangian strain tensor
E
and the Eulerian strain tensor
e
are expressed in
terms of
C
and
c
by the following formulas that follow from (
11.15
), and the
definitions of
C
and
c
given as (
11.27
) and (
11.28
) above,
2E
¼
C
1
;
2e
¼
1
c
:
(11.29)
The eigenvalues of the various strain measures may be interpreted using the
concept of
stretch
. The stretch
l
(
N
)
in the fiber coincident with d
X
is defined by
r
dx
dx
l
ðNÞ
¼
;
(11.30)
dX
dX
where
N
is a unit vector in the direction of d
X
. The related concept of extension
d
(
N
)
is then defined in terms of stretch by
d
ðNÞ
¼ l
ðNÞ
1
:
(11.31)
As a illustration of homogeneous deformations, and of the relationship of the
stretch concept to the various strain measures that have been introduced, the special
case of
pure homogeneous deformation
s is considered. A pure homogeneous
deformation is a deformation for which the rotation
R
¼
1
and the deformation
gradient tensor become symmetric,
F
V
. In its principal coordinate system
the deformation gradients of a pure homogeneous deformation have the
representation
¼
U
¼
2
4
3
5
;
l
I
00
0
F
¼
U
¼
V
¼
l
II
0
(11.32)
00
l
III
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