Modeling Material Symmetry - Continuum Mechanics of Anisotropic Materials - page 84

as may be verified directly. The reflective transformation in six dimensions,
denoted by R ðaÞ , is constructed from R ðaÞ using (A167), thus

p

2 2

R ðaÞ

¼

2
4

3
5

2

ð 1 2 a 1 Þ 2 a 1 a 2 2 a 1 a 3

2 2

p

p p

4 a 1 a 2 a 3 2 a 1 a 3 ð 2 a 1 1 Þ 2 a 1 a 2 ð 2 a 1 1 Þ

2

2 a 1 a 2 ð 1 2 a 2 Þ 2 a 2 a 3

p 2 2

p

p

2 a 2 a 3 ð 2 a 2 1 Þ 4 a 1 a 2 a 3 2 a 1 a 2 ð 2 a 2 1 Þ

2

2 a 1 a 3 2 a 2 a 3 ð 1 2 a 3 Þ

p p 2 p

2 a 2 a 3 ð 2 a 3 1 Þ 2 a 1 a 3 ð 2 a 3 1 Þ 4 a 1 a 2 a 3

2 a 1 þ 8 a 2 a 3 1
2

2 a 1 a 2 ð 4 a 3 1 Þ
2

2 a 1 a 3 ð 4 a 2 1 Þ
2 2

4 a 1 a 2 a 3 2 a 2 a 3 ð 2 a 2 1 Þ 2 a 2 a 3 ð 2 a 3 1 Þ

p p

p

2 a 1 a 2 ð 4 a 3 1 Þ
2

2 a 2 þ 8 a 1 a 3 1
2

2 a 2 a 3 ð 4 a 1 1 Þ
2 2

2 a 1 a 3 ð 2 a 1 1 Þ 4 a 1 a 2 a 3 2 a 1 a 3 ð 2 a 3 1 Þ

p p

p

2 a 1 a 3 ð 4 a 2 1 Þ
2

2 a 2 a 3 ð 4 a 1 1 Þ
2

2 a 3 þ 8 a 2 a 1 1
2 2

2 a 1 a 2 ð 2 a 1 1 Þ 2 a 1 a 2 ð 2 a 2 1 Þ 4 a 1 a 2 a 3

p p

p

(4.3)

As an example of the application of the result ( 4.3 ), the six-dimensional
transformations corresponding to planes of mirror symmetry in the e 1 and e 2
directions,

2
4

3
5 ; R ðe 2 Þ ¼

2
4

3
5 ;

100
010
001

100
0 R ðe 1 Þ ¼ 10

001

(4.4)

respectively, are given by

2
4

3
5

2
4

3
5

10000 0
01000 0
00100 0
00010 0
0000

100000
010000
001000
000

R ðe 1 Þ

; R ðe 2 Þ

¼ ¼ ; (4.5)

100
000010
00000

10
00000 1 1

respectively. Other examples are the cases when the normals to the plane of
reflective symmetry are vectors in the e 1 , e 2 plane, a ¼ cos y e 1 þ sin y e 2 ,or

e 3 . In these cases R ðy 12 Þ and R ðy 23 Þ are given by

the e 2 , e 3 plane, a ¼ cos y e 2 þ sin y

2
4

3
5 ;

2
4

3
5 ;

cos 2 y sin 2 y 0 1 0 0

R ðy 12 Þ ¼ R ðy 23 Þ ¼ sin 2 y cos 2 y 0 0 cos 2 y sin 2 y (4.6)

0 0 1 0 sin 2 y cos 2 y