Page 50 - Introduction to Continuum Mechanics

PartB Symmetric and Antisymmetric Tensors 35

This is the transformation law for the components of a vector. Thus, fy are components of a
vector.
Another example which will be important later when we discuss the relationship between
stress and strain for an elastic body is the following: If 7^- and EJJ are components of arbitrary
second order tensors T and E then
T
ij = CijklEkl
for all coordinates, then C^ are components of a fourth order tensor. The proof for this
example follows that of the previous example.

2B15 Symmetric and Antisymmetric Tensors
7*
A tensor is said to be symmetric if T = T . Thus, the components of a symmetric tensor
have the property,

i.e.,

r
A tensor is said to be antisymmetic if T = -T . Thus, the components of an antisymmetric
tensor have the property

i.e.,

and

Any tensor T can always be decomposed into the sum of a symmetric tensor and an
antisymmetric tensor. In fact,

where

and

It is not difficult to prove that the decomposition is unique (see Prob. 2B27)