Page 46 - Introduction to Continuum Mechanics

Part B Transformation Law for Cartesian Components of a Tensor 31

We can also express the unprimed components in terms of the primed components. Indeed,
premultiply Eq. (2B13.1c) with [Q] and postmultiply it with [Q] , we obtain, since
[QMQf=[Qf[Q] = m>

Using indicia! notation, Eq. (2B13.2a) reads

Equations (2B13.1& 2B13.2) are the transformation laws relating the components of the
same tensor with respect to different Cartesian unit bases. It is important to note that in these
equations, [T] and [TJ'are different matrices of the same tensor T. We note that the equation
T r
[T]' = [Q] [T][Q] differs from the equation T' = Q TQ in that the former relates the com-
ponents of the same tensor T whereas the latter relates the two different tensors T and T '.

Example 2B 13.1
Given the matrix of a tensor T in respect to the basis {e/}:

"0 1 0~
[T] = 12 0
L° 0 1
Find [T] e:, i.e., find the matrix of T with respect to the {e/} basis, where {e/} is obtained by
rotating {e/} about €3 through 90°. (see Fig. 2B.5).
Solution. Since ei — 62,62 = -ej and 63 = 63, by Eq. (2Bll.lb), we have

0 -1 0"
[Q]= 1 0 0
0 0 1
Thus, Eq.(2B13.1c) gives
" 0 1 O] [0 1 0] |~0 -1 Ol [ 2 -1 0"
[T]' =-10 0 12 0 1 00 = - 1 0 0
[ 0 0 IJ [0 0 Ij [0 0 IJ [ 0 0 1

i.e., TU = 2, T{ 2 = -1, T{ 3 = 0,r 2 '! = -1, etc.

Example 2B13.2

Given a tensor T and its components Tjy and Tjj with respect to two sets of bases {e/} and
{e/ }. Show that 7}/ is invariant with respect to this change of bases, i.e., 7}/ = 7//.