Page 40 - Introduction to Continuum Mechanics

Part B Orthogonal Tensor 25

1 r
This means that Q~ =Q and from Eq. (2B9.3),

In matrix notation, Eqs. (2B10.2a) take the form:

and in subscript notation, these equations take the form:

Example 2B 10.1

The tensor given in Example 2B2.2, being a reflection, is obviously an orthogonal tensor.
Verify that [T][T] = [I] for the [T] in that example. Also, find the determinant of [T].
Solution. Using the matrix of Example 2B7.1:
f-1 0 0] [-1 0 0] fl 0 0"
r
[T][T] = 01 0 010=01 0
[ 0 0 1J [ 0 0 1J [00 1
The determinant of [T] is
-10 0
|T| = 0 1 0 = -1
0 0 1

Example 2B 10.2

The tensor given in Example 2B2.3, being a rigid body rotation, is obviously an orthogonal
tensor. Verify that [R][R] = [I] for the [R] in that example. Also find the determinant of [R].
Solution. It is clear that

[cos0 -sin<9 ol f cos0 sin0 o] |"l 0 0*
r
[R][R] = sin0 cos0 0 -sin0 cos0 0=01 0
[ 0 0 ll [ 0 0 ij [0 0 1
cos# -sin# 0
det[R]s|R| = sin0 cos0 0 =+ 1
0 0 1

The determinant of the matrix of any orthogonal tensor Q is easily shown to be equal to
either + 1 or -1. In fact,

[QHQf =[i]