Page 41 - Introduction to Continuum Mechanics

26 Tensors

therefore,

From the previous examples we can see that the value of 4-1 corresponds to rotation and -1
corresponds to reflection.

2B11 Transformation Matrix Between Two Rectangular Cartesian Coordinate
Systems.

Suppose {e,-} and {ej} are unit vectors corresponding to two rectangular Cartesian coor-
dinate systems (see Fig. 2B.3). It is clear that {e,-} can be made to coincide with {e^ } through
either a rigid body rotation (if both bases are same handed) or a rotation followed by a
:
reflection (if different handed). That is {e/} and {e,} can be related by an orthogonal tensor
Q through the equations

i.e.,

where

or

We note that Qn - e^-Qej = ej-ei = cosine of the angle between *i and ei,
e = e
e =
012 = e i" Q 2 i' 2 cosine of the angle between ej and 63, etc. In general, Qij = cosine
:
of the angle between e,- and e. which may be written:

The matrix of these directional cosines, i.e., the matrix

"011 012 013
[Q]= 021 022 023
031 032 033