Page 228 - Intro to Tensor Calculus

222

Figure 2.3-11. Rotation of unit square

Transformation of an Arbitrary Element

In two dimensions, we consider a rectangular element ABCD as illustrated in the figure 2.3-12.
Let the points ABCD have the coordinates

A(x, y), B(x +∆x, y), C(x, y +∆y), D(x +∆x, y +∆y) (2.3.29)

and denote by
u = u(x, y), v = v(x, y)
the displacement field associated with each of the points in the material continuum when it undergoes a
deformation. Assume that the deformation of the element ABCD in figure 2.3-12 can be represented by the
matrix equation

x b 11 b 12 x
= (2.3.30)
y b 21 b 22 y
where the coefficients b ij ,i,j =1, 2, 3 are to be determined. Let us define u = u(x, y) as the horizontal
displacement of the point (x, y)and v = v(x, y) as the vertical displacement of the same point. We can now
express the displacement of each of the points A, B, C and D in terms of the displacement field u = u(x, y)
and v = v(x, y). Consider first the displacement of the point A to A . Here the coordinates (x, y) deform to
0
the new coordinates
x = x + u, y = y + v.

That is, the coefficients b ij must be chosen such that the equation

x + u b 11 b 12 x
= (2.3.31)
y + v b 21 b 22 y
0
is satisfied. We next examine the displacement of the point B to B . This displacement is described by the
coordinates (x +∆x, y) transforming to (x, y), where

x = x +∆x + u(x +∆x, y), y = y + v(x +∆x, y). (2.3.32)