Biomedical Engineering Reference
In-Depth Information
coordinates, was removed by spatial or material differentiation. It has also been
noted that the special case of
F
1
corresponds to no rotational and no deforma-
tional motion. Further it has been shown (
2.50
) that
Y
is associated with pure rigid
object rotation. This means that
E
must be the tensor representing the deformation.
This is indeed the case, as will be shown below.
E
is called the
infinitesimal strain
tensor
and
Y
is called the
infinitesimal rotation tensor
. The representation (
2.51
) for
the tensor of deformation gradients then demonstrates that, for infinitesimal
motions,
F
¼
1
may be decomposed into the sum of two terms,
E
and
Y
, which
represent the deformational and rigid rotational characteristics of the infinitesimal
motion, respectively.
The strain tensor
E
, defined by the first of
(
2.49
), has the component
representation
2
4
3
5
2
@
u
1
@
u
1
x
2
þ
@
u
2
@
u
1
x
3
þ
@
u
3
@
x
1
@
@
x
1
@
@
x
1
1
2
@
u
j
x
i
þ
@
u
i
1
2
@
u
1
x
2
þ
@
u
2
2
@
u
2
@
u
2
x
3
þ
@
u
3
E
¼
¼
:
(2.52)
@
@
x
j
@
@
x
1
@
x
2
@
@
x
2
@
u
1
x
3
þ
@
u
3
@
u
2
x
3
þ
@
u
3
2
@
u
3
@
@
x
1
@
@
x
2
@
x
3
The components of
E
along the diagonal are called normal strains and the
components off the diagonal are called shear strains. The normal strains,
E
11
,
E
22
,
and
E
33
, are measures of change in length per unit length along the 1, 2 and 3 axes,
respectively, and the shear strains,
E
23
,
E
13
, and
E
12
, are one-half of the changes in
the angle between the 2 and 3 axes, the 1 and 3 axes and the 1 and 2 axes,
respectively.
The geometric interpretation of the components of the strain tensor
E
stated in
the previous paragraph will be analytically developed here. Let
dx
1
be a vector of
infinitesimal length representing the present position of an infinitesimal material
filament coinciding with the
x
1
at time
t
. The displacement of this material filament
instantaneously coincident with
dx
1
is
du
1
¼
E
11
dx
1
, a result that follows from the
entry in the first column and first row of (
2.52
). The expression
du
1
¼
E
11
dx
1
is the
change in length of
dx
1
as a consequence of the strain as illustrated in Fig.
2.8
. Thus
the geometric interpretation of
E
11
¼
(
du
1
/
dx
1
) is that it is the change in length per
unit length of
dx
1
. Similar geometric interpretations exist for
E
22
and
E
33
.
The geometric interpretation of the normal strain components
E
11
,
E
22
and
E
33
is
easily extended to obtain a geometric interpretation of the trace of the small strain
tensor tr
E
, or equivalently the divergence of the displacement field
∇
u
,tr
E
¼
∇
dx
1
dx
2
dx
3
represents an undeformed element of volume (Fig.
2.9
),
the deformed volume is given by
dv
u
.If
dv
o
¼
¼
(
dx
1
þ
du
1
)(
dx
2
þ
du
2
)(
dx
3
þ
du
3
). Using
du
1
¼
E
11
dx
1
,
du
2
¼
E
22
dx
2
and
du
3
¼
E
33
dx
3
, the deformed volume is given by
dv
¼
(1
þ
E
11
)(1
þ
E
22
)(1
þ
E
33
)
dv
o
.
Expanding
dv
¼
(1
þ
E
11
)(1
þ
E
22
)
(1
þ
E
33
)
dv
o
and recognizing that the squares of displacement gradients (
2.37
)
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