Page 227 - Intro to Tensor Calculus

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Figure 2.3-9. Superposition of shearing elements

Figure 2.3-10. Rotation of element P 1 P 2

where α is very small. Applying this transformation to the unit square previously considered we obtain the
results illustrated in the figure 2.3-10.
Note the difference in the direction of shearing associated with the transformation equations (2.3.27)
and (2.3.23) illustrated in the figures 2.3-6 and 2.3-10. If the matrices appearing in the equations (2.3.24)
and (2.3.27) are multiplied and we neglect product terms because α is assumed to be very small, we obtain
the matrix equation

x 1 α x 1 0 x 0 α x
= = + . (2.3.28)
y −α 1 y 0 1 y −α 0 y
| {z } | {z }
identity rotation
This can be interpreted as a superposition of the transformation equations (2.3.24) and (2.3.27) which
represents a rotation of the unit square as illustrated in the figure 2.3-11.
The matrix on the right-hand side of equation (2.3.28) is referred to as a rotation matrix. The ideas
illustrated by the above simple transformations will appear again when we consider the transformation of an
arbitrary small element in a continuum when it under goes a strain. In particular, we will be interested in
extracting the rigid body rotation from a deformed element and treating this rotation separately from the
strain displacement.