Page 35 - Introduction to Continuum Mechanics

20 Tensors

(c)Since S(Ra) = (SR)a, the resultant rotation is given by the single transformation SR
whose components are given by the matrix
"l 0 Ol [0 -1 0] |~0 -1 0"
[SR]= 0 0 -1 1 00= 0 0- 1
[0 1 OJ [0 0 IJ [l 0 0
(d)In a manner similar to (c) the resultant rotation is given by the single transformation RS
whose components are given by the matrix
"o -i ol fi o ol [bo i"
[RS]= 1 0 0 00- 1 = 10 0
[0 0 IJ [0 1 OJ [0 1 0
(e)Let r be the initial position of the point P. Let r* and r** be the rotated position of P
after the rotations of part (c) and part (d) respectively. Then
[o ~i ol [i] F-i"
[r*] = [SR][r] = 00-11 = 0
1 0 OJ I OJ [ 1
i.e.,
r* = -e!+e 3
and
fo o il [i] [o"
[r**] = [RS][r] =10 0 1 = 1
l
L° °J H L 1
i.e.,
**
r = 62+63
This example further illustrates that the order of rotations is important.

286 Transpose of a Tensor

r
The transpose of a tensor T, denoted by T , is defined to be the tensor which satisfies the
following identity for all vectors a and b;

T
It can be easily seen that T is a tensor. From the above definition, we have

Thus,

or