Page 299 - Rock Mechanics For Underground Mining

ENERGY TRANSMISSION IN ROCK
Figure 10.8 Specification of plane
waves propagating in the x co-
ordinate direction.
A case of particular interest occurs for a bar with a free end, i.e. a composite bar
in which 2 = C 2 = 0. Then n = 0, and equations 10.19 and 10.21 yield
r =− 0 (10.23)
V r = V 0 (10.24)
That is, a compressive pulse is reflected completely as a tensile pulse, while the
sense of particle motion in the reflected pulse is in the original (forward) direction
of pulse propagation. The generation of a tensile stress at a free face by reflection
of a compressive pulse provides a plausible mechanism for development of slabs or
spalls at a surface during rock blasting. The issue has been discussed in detail by Hino
(1956), among others.
10.3.2 Plane waves in a three-dimensional medium
In the following discussion, a wave is assumed to be propagating in the x co-ordinate
direction in a three-dimensional, elastic isotropic continuum. Passage of the wave
induces transient displacements u x (t), u y (t), u z (t) at any point in the medium as
indicated in Figure 10.8. The essential notion in the concept of a plane wave is that, at
any instant in time, displacements at all points in a particular yz plane are identical,
i.e. (u x , u y , u z ) are independent of (y, z). Alternatively, the definition of a plane wave
may be expressed in the form
u x = u x (x), u y = u y (x), u z = u z (x) (10.25)
The derivation of the differential equations of motion for the components of a plane
wave proceeds in a manner analogous to that for the longitudinal bar wave. From the
general strain–displacement relations given in equations 2.35 and 2.36, the transient
strains associated with a plane wave are obtained from equation 10.25 as
ε xx =−∂u x /∂x, ε yy = ε zz = 0 (10.26)
xy =−∂u y /∂x, yz = 0, zx =−∂u z /∂x (10.27)
For the case of axisymmetric uniaxial normal strain defined by equation 10.26, the
equations of isotropic elasticity yield
yy = zz = /(1 − ) xx
Substitution of these expressions in the equation defining the x component of normal
strain, i.e.
ε xx = 1/E[ xx − ( yy + zz )]
yields, after some manipulation
ε xx = [(0.5 − )/(1 − )] xx /G (10.28)
For the shear strain components, Hooke’s Law gives
xy = 1/G xy , zx = 1/G zx (10.29)
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