Page 49 - Rock Mechanics For Underground Mining

DISPLACEMENT AND STRAIN

and
du x =− z dy
(2.28)
du y = z dx

The total displacement due to the various rigid-body rotations is obtained by addi-
tion of equations 2.26, 2.27 and 2.28, i.e.

du x =− z dy + y dz
du y = z dx − x dz
du z =− y dx + x dy

These equations may be written in the form
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
du x 0 − z y dx
⎣ du y ⎦ = ⎣ z 0 − x ⎦ ⎣ dy ⎦ (2.29a)
du z − z x 0 dz
or

[d ] = [Ω][dr] (2.29b)

The contribution of deformation to the relative displacement [d ] is determined
by considering elongation and distortion of the element. Figure 2.8 represents the
Figure 2.8 Displacement compo- elongation of the block in the x direction. The element of length dx is assumed to be
nents produced by pure longitudinal
strain. homogeneously strained, in extension, and the normal strain component is therefore
defined by

du x
ε xx =
dx
Considering the y and z components of elongation of the element in a similar way,
gives the components of relative displacement due to normal strain as

du x = ε xx dx
du y = ε yy dy (2.30)
du z = ε zz dz

The components of relative displacement arising from distortion of the element
are derived by considering an element subject to various modes of pure shear strain.
Figure 2.9 shows such an element strained in the x, y plane. Since the angle is
small, pure shear of the element results in the displacement components
du x = dy
du y = dx

Since shear strain magnitude is defined by

Figure 2.9 Displacement produced
by pure shear strain. xy = − = 2
2
31