Page 97 - Introduction to Continuum Mechanics

82 Kinematics of a Continuum

so that the whole material line OA is motionless.
On the other hand, the material coordinates for the material line CB are

so that according Eq. (ii)

In other words, the material line has moved horizontally through a distance of kt (see Fig. 3.2).
The material coordinates for the material line OC are (Xi,X 2,X$) - (Q^G), so tnat f° r
e
the particles along this line (xi,x 2,x$) = (ktX 2^ 2,Q\ Th fact that xi~ktX 2 means that the
straight material line OC remains a straight line OC 'at time t as shown in Fig. 3.2, The situation
for the material limAB is similar. Thus, at time t, the side view of the cube changes from that
of a square to a parallelogram as shown. Since x$ = X$ at all time for all particles, it is clear
that all motions are parallel to the plane x 3 = 0. The motion given in this example is known
as simple shearing motion.

Example 3.1.2
Let

Express the simple shearing motion given in Example 3.1.1 in terms of (Y\, Y 2, Y%)
Solution. Straight forward substitutions give

These equations, i.e.,

obviously also describe the simple shearing motion just as the equations given in the previous
example. The triples (Yi,Y 2,Y$) are a ^° material coordinates in that they also identify the
particles in the continuum although they are not the coordinates of the particles at any time.
This example demonstrates the fact that while the positions of the particles at some reference
time t 0 can be used as the material coordinates, the material coordinates need not be the
positions of the particle at any time. However, within this book, all material coordinates will
be coordinates of the particles at some reference time.