Page 51 - Introduction to Continuum Mechanics

36 Tensors

Example 2B15.1

Show that if T is symmetric and W is antisymmetric, then tr(TW)=0.
Solution. We have, [see Example 2B8.4]

T T
Since T is symmetric and W is antisymmetric, therefore, by definition, T=T , W= — W . Thus,
(see Example 2B8.1)

Consequently, 2tr(TW)=0. That is,

2B16 The Dual Vector of an Antisymmetric Tensor

The diagonal elements of an antisymmetric tensor are always zero, and, of the six non-
=
diagonal elements, only three are independent, because T^i ~ ~^12>^13 ~^3i
and T23 = ~ r 32 . Thus, an antisymmetric tensor has really only three components, just like a
vector. Indeed, it does behavior like a vector. More specifically, for every antisymmetric tensor
4
T, there corresponds a vector f , such that for every vector a the transformed vector, Ta, can
4
be obtained from the cross product of t with a. That is,

This vector, i , is called the dual vector (or axial vector ) of the antisymmetric tensor. The
form of the dual vector is given below:
From Eq.(2B16.1), we have, since a-bxc = b-cxa,

4
4
4
7i2 = e 1-Te 2 = e 1 -f Xe 2 = t -e 2 Xe 1 = -f -e 3 = -f$
4
4
T 31 = e 3-T Cl = e 3 -f xe 1 = t^xes = -f ^ = -$
^23 = * 2-Te 3 = e^f^ = <*-e 3xe 2 = -f-^ = -$
Similar derivations will give T 21 = 1$, T 13 = /2,T 32 = fi and T\\ = T 22 = T 33 = 0. Thus, wi/y
an antisymmetric tensor has a dual vector defined by Eq.(2B16.1). It is given by:

or, in indicial notation