Page 38 - Introduction to Continuum Mechanics

Part B Identity Tensor and Tensor Inverse 23

2B9 identity Tensor and Tensor Inverse

The linear transformation which transforms every vector into itself is called an identity
tensor. Denoting this special tensor by I, we have, for any vector a,

and in particular,

Thus, the components of the identity tensor are:

i.e.,

It is obvious that the identity matrix is the matrix of I for all rectangular Cartesian coordinates
and that TI = IT = T for any tensor T. We also note that if Ta = a for any arbitrary a, then
T = I.

Example 2B9.1

Write the tensor T, defined by the equation Ta = A:a, where k is a constant and a is arbitrary,
in terms of the identity tensor and find its components.
Solution. Using Eq. (2B9.1) we can write A; a as fcla so that Ta = fca becomes
Ta = Ma
and since a is arbitrary
T = fcl
The components of this tensor are clearly,
T • = kd-
1 fj
KVy

Given a tensor T, if a tensor S exists such that ST=I then we call S the inverse of T or
1
-1
S=T . (Note: With T~ T=T~ 1+1 =T°=I, the zeroth power of a tensor is the identity
tensor). To find the components of the inverse of a tensor T is to find the inverse of the matrix
of T. From the study of matrices we know that the inverse exists as long as detT^O (that is, T