Page 34 - Introduction to Continuum Mechanics

PartB Product of Two Tensors 19

and
(TS)(Va) = T(S(Va))
i.e.,

Thus, the tensor product is associative. It is, therefore, natural to define the integral positive
powers of a transformation by these simple products, so that

Example 2B5.1
(a)Let R correspond to a 90° right-hand rigid body rotation about the^-axis. Find the matrix
ofR.
(b)Let S correspond to a 90°right-hand rigid body rotation about thejcj-axis. Find the matrix
ofS.
(c)Find the matrix of the tensor that corresponds to the rotation (a) then (b).
(d)Find the matrix of the tensor that corresponds to the rotation (b) then (a).
(e)Consider a point P whose initial coordinates are (1,1,0). Find the new position of this
point after the rotations of part (c). Also find the new position of this point after the rotations
of part (d).
Solution, (a) For this rotation the transformation of the base vectors is given by

Rej = e 2
Re 2 = -ej

Re3 = e 3
so that,
0 -1 0~
[R]= 1 0 0
00 1
(b)In a similar manner to (a) the transformation of the base vectors is given by

Se 1 = e 1
Se 2 = e 3
Se 3 = -e 2
so that,
"l 0 0"
[S]= 0 0 -1
[0 1 0