This preview shows pages 1–3. Sign up to view the full content.
Section 1.8
Solid Mechanics Part III
Kelly
67
1.8 Tensors
Here the concept of the
tensor
is introduced.
Tensors can be of different
orders
– zeroth
order tensors, firstorder tensors, secondorder tensors, and so on.
Apart from the zeroth
and first order tensors (see below), the secondorder tensors are the most important
tensors from a practical point of view, being important quantities in, amongst other topics,
continuum mechanics, relativity, electromagnetism and quantum theory.
1.8.1
Zeroth and First Order Tensors
A
tensor of order zero
is simply another name for a scalar
α
.
A
firstorder tensor
is simply another name for a vector
u
.
1.8.2
Second Order Tensors
Notation
Vectors:
lowercase boldface Latin letters, e.g.
a
,
r
,
q
2
nd
order Tensors:
uppercase boldface Latin letters, e.g.
F
,
T
,
S
Tensors as Linear Operators
A
second
order tensor
T
may be
defined
as an operator that acts on a vector
u
generating
another vector
v
, so that
v
u
T
=
)
(,
o
r
1
v
Tu
v
u
T
=
=
⋅
or
Secondorder Tensor
(1.8.1)
The secondorder tensor
T
is a
linear operator
(or
linear transformation
)
2
, which
means that
()
Tb
Ta
b
a
T
+
=
+
…
distributive
( )
Ta
a
T
=
…
associative
This linearity can be viewed geometrically as in Fig. 1.8.1.
Note:
•
the vector may also be defined in this way, as a mapping
u
that acts on a vector
v
, this time
generating a scalar
α
,
=
⋅
v
u
.
This transformation (the dot product) is linear (see properties
(2,3) in §1.1.4).
Thus a firstorder tensor (vector) maps a firstorder tensor into a zerothorder
tensor (scalar), whereas a secondorder tensor maps a firstorder tensor into a firstorder tensor.
It will be seen that a thirdorder tensor maps a firstorder tensor into a secondorder tensor, and
so on
1
both these notations for the tensor operation are used; here, the convention of omitting the “dot” will be
used
2
An operator or transformation is a special function which maps elements of one type into elements of a
similar type; here, vectors into vectors
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentSection 1.8
Solid Mechanics Part III
Kelly
68
Figure 1.8.1: Linearity of the second order tensor
Further, two tensors
T
and
S
are said to be equal if and only if
Tv
Sv
=
for all vectors
v
.
Example (of a Tensor)
Suppose that
F
is an operator which transforms every vector into its mirrorimage with
respect to a given plane, Fig. 1.8.2.
F
transforms a vector into another vector and the
transformation is linear, as can be seen geometrically from the figure.
Thus
F
is a
secondorder tensor.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '11
 Staff

Click to edit the document details