Page 226 - Intro to Tensor Calculus

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Figure 2.3-7. Shearing parallel to the x axis

Figure 2.3-8. Shearing parallel to x and y axes

Again, if α is very small, we may use the approximation tan α ≈ α and interpret α as an angular rotation
of the element P 1 P 4 to the position P P . Now let us multiply the matrices given in equations (2.3.23) and
0
0
1 4
(2.3.24). Note that the order of multiplication is important as can be seen by an examination of the products

x 1 0 1 α x 1 α x
= =
y β 1 0 1 y β 1+ αβ y
(2.3.25)

x 1 α 1 0 x 1+ αβ α x
= = .
y 0 1 β 1 y β 1 y
In equation (2.3.25) we will assume that the product αβ is very, very small and can be neglected. Then the
order of matrix multiplication will be immaterial and the transformation equation (2.3.25) will reduce to

x 1 α x
= . (2.3.26)
y β 1 y
Applying this transformation to our unit square we obtain the simultaneous shearing parallel to both the x
and y axes as illustrated in the figure 2.3-8.
This transformation can then be interpreted as the superposition of the two shearing elements depicted
in the figure 2.3-9.
For comparison, we consider also the transformation equation

x 1 0 x
= (2.3.27)
y −α 1 y