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Section 2.8
Solid Mechanics Part III
Kelly
263
2.8 Objectivity and Objective Tensors
2.8.1
Dependence on Observer
Consider a rectangular block of material resting on a circular table.
A person stands and
observes the material deform, Fig. 2.8.1a.
The dashed lines indicate the undeformed
material whereas the solid line indicates the current state. A second observer is standing
just behind the first, but on a step ladder – this observer sees the material as in 2.8.1b.
A
third observer is standing around the table,
o
45 from the first, and sees the material as in
Fig. 2.8.1c.
The deformation can be described by each observer using concepts like displacement,
velocity, strain and so on.
.
However, it is clear that the three observers will in general
record different values for these measures, since their perspectives differ.
The goal in what follows is to determine which of the kinematical tensors are in fact
independent
of observer.
Since the laws of physics describing the response of a
deforming material must be independent of any observer, it is these particular tensors
which will be more readily used in expressions to describe material response.
Figure 2.8.1: a deforming material as seen by different observers
Note that Fig. 2.8.1 can be interpreted in another, equivalent, way.
One can imagine
one
static observer, but this time with the material moved into three different positions.
This
viewpoint will be returned to in the next section.
2.8.2
Change of Reference Frame
Consider two
frames of reference
, the first consisting of the origin
o
and the basis
{}
i
e
,
the second consisting of the origin
*
o
and the basis
{ }
*
i
e
, Fig. 2.8.2.
A point
x
in space is
then identified as having position vector
i
i
x
e
x
=
in the first frame and position vector
*
*
*
i
i
x
e
x
=
in the second frame.
When the origins
o
and
*
o
coincide,
*
x
x
=
and the vector components
i
x
and
*
i
x
are
related through Eqn. 1.5.3,
*
j
ij
i
x
Q
x
=
, or
i
j
ij
i
i
x
Q
x
e
e
x
*
=
=
, where
[ ]
Q
is the
(a)
(b)
(c)
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View Full Document Section 2.8
Solid Mechanics Part III
Kelly
264
transformation matrix 1.5.4,
*
j
i
ij
Q
e
e
⋅
=
.
Alternatively, one has Eqn. 1.5.5,
j
ji
i
x
Q
x
=
*
,
or
*
*
*
*
i
j
ji
i
i
x
Q
x
e
e
x
=
=
.
Figure 2.8.2: two frames of reference
With the shift in origin
*
o
o
a
−
=
, one has
*
*
*
*
*
*
i
i
i
j
ji
i
i
a
x
Q
x
e
e
e
x
+
=
=
(2.8.1)
where
*
*
i
i
a
e
a
=
.
Alternatively,
i
i
i
j
ij
i
i
a
x
Q
x
e
e
e
x
−
=
=
*
(2.8.2)
where
i
i
a
e
a
=
, with
j
ji
i
a
Q
a
=
*
.
Formulae 2.8.12 relate the coordinates of the position vector to a point in space as
observed from one frame of reference to the coordinates of the position vector to the
same
point as observed from a different frame of reference.
Finally, consider the position vector
x
, which is defined relative to the frame
()
i
e
o
,.
T
o
an observer in the frame
( )
*
*
,
i
e
o
, the
same
position vector would appear as
*
x
, Fig.
2.8.3.
Rotating this vector
*
x
through
T
Q
(the tensor which rotates the basis
{ }
*
i
e
into
the basis
{}
i
e
) and adding the vector
a
then produces
*
x
:
( )
a
x
Q
x
+
=
*
T
*
(2.8.3)
This relation will be discussed further below.
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.
 Fall '11
 Staff

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