Page 30 - Introduction to Continuum Mechanics

Part B Components of a Tensor 15

"-1 0 0"
[T] = 01 0
0 0 1
J Cj
We note that this is only one of the infinitely many matrices of the tensor T, each depends
on a particular choice of base vectors. In the above matrix, the choice of e, is indicated at the
bottom right corner of the matrix. If we choose ei and 62 to be on a plane perpendicular to
the mirror with each making 45° with the mirror as shown in Fig. 2B.1 and 63 points straight
out from the paper. Then we have
Tel = «2

Te 2 = ej
Te3 = e 3
Thus, with respect to {e/}, the matrix of the tensor is

0 1 0"
[T]' =10 0
001 -
L Jej

Throughout this book, we shall denote the matrix of a tensor T with respect to the basis
e,- by either [T] or [Tjy] and with respect to the basis e/' by either [T]' or[7)y] The last
two matrices should not be confused with [T'], which represents the matrix of the tensor
T ' with respect to the basis e,-.

Example 2B2.3
Let R correspond to a right-hand rotation of a rigid body about the *3-axis by an angle B.
Find a matrix of R.
Solution. From Fig. 2B.2 it is clear that
Rej = cos0ei+sin#e2
Re2 = -sin0ei+cos0e2
Re 3 = e 3
Thus,
cos# -sin# 0
[R] = sinfl cos0 0
0 0 1
L Jcj