Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 353

A.6.4. Is it possible for an open product of vectors to be an orthogonal matrix?
A.6.5. Transform the components of the vector v (L) ¼ [1, 2, 3] to a new (the

Greek) coordinate system using the transformation

2 q

q

2 3

1

2

4 5

1
2

p

Q ¼

1 1

p p

p 2 p 2

0

A.7 Second Order Tensors

Scalars are tensors of order zero; vectors are tensors of order one. Tensors of order
two will be defined using vectors. For brevity, we shall refer to “tensors of order two”
simply as “tensors” throughout most of this section. For application in physical
theories, physicists generated the notion of a tensor, very similar to the notion of
a vector, but generalizing the vector concept. In classical dynamics the essential
concepts of force, velocity, and acceleration are all vectors; hence the mathe-
matical language of classical dynamics is that of vectors. In the mechanics of
deformable media the essential concepts of stress, strain, rate of deformation,
etc. are all second order tensors, thus, by analogy, one can expect to deal quite
frequently with second order tensors in this branch of mechanics. The reason for
this widespread use of tensors is that they enjoy, like vectors, the property of
being invariant with respect to the basis, or frame of reference, chosen.
The definition of a tensor is motivated by a consideration of the open or dyadic
product of the vectors r and t . Recall that the square matrix formed from r and t is
called the open product of the n-tuples r and t , it is denoted by r t , and defined by

(A.30) for n-tuples. We employ this same formula to define the open product of the
vectors r and t . Both of these vectors have representations relative to all bases in the
vector space, in particular the Latin and the Greek bases, thus from (A.72)

r ¼ r i e i ¼ r a e a ; t ¼ t j e j ¼ t b e b : (A.78)

The open product of the vectors r and t , r t , then has the representation

r t ¼ r i t j e i e j ¼ r a t b e a e b : (A.79)

This is a special type of tensor, but it is referred to the general second order
tensor basis, e i e j ,or e a e b . A general second order tensor is the quantity
T defined by the formula relative to the bases e i e j , e a e b and, by implication,
any basis in the vector space: