Page 300 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

Equations 10.28 and 10.29, on rearrangement and introduction of the strain–
displacement relations (equations 10.26 and 27), reduce to

xx =−[(1 − )/(0.5 − )]G∂u x /∂x (10.30)
xy =−G∂u y /∂x, zx =−G∂u z /∂x (10.31)

In formulating governing equations for wave propagation, the requirement is to
account for the inertial force associated with passage of the wave. Consider the small
element of the body shown in Figure 10.9. If the net x-component of force on the
body is X per unit volume, introduction of the d’Alembert force ¨ u x , in the sense
opposing the net force, produces the pseudo-equilibrium condition described by the
equation
Figure 10.9 Force and stress com-
ponents acting on an elementary free X + ¨ u x = 0
body subject to transient motion in the
x co-ordinate direction.
or
X =− ¨ u x (10.32)
Similar expressions can be established for the other co-ordinate directions.
For the geomechanics convention for sense of positive stresses being used here, the
differential equations of equilibrium (equations 2.21), when combined with equation
10.32 (and similar equations for the other co-ordinate directions) become

2
∂ xx /∂x + ∂ xy /∂y + ∂ zx /∂z = X =− ∂ u x /∂t 2
2
∂ xy /∂x + ∂ yy /∂y + ∂ yz /∂z = Y =− ∂ u y /∂t 2 (10.33)
2
∂ zx /∂x + ∂ yz /∂y + ∂ zz /∂z = Z =− ∂ u z /∂t 2
The definition of the plane wave, and equations 10.30 and 10.31, reduce equations
10.33 to

2 2
2
2
2
2
∂ u x /∂t = [(1 − )/(0.5 − )](G/ )(∂ u x /∂x ) = C ∂ u x /∂x 2 (10.34)
p
2
2 2
2
2
2
∂ u y /∂t = (G/ )∂ u y /∂t = C ∂ u y /∂x 2
s
(10.35)
2 2 2 2 2 2 2
∂ u z /∂t = (G/ )∂ u z /∂t = C ∂ u z /∂x
s
where
1
C p ={[(1 − )/(0.5 − )](G/ )} 2
1
C s = (G/ ) 2
Equations 10.34 and 10.35 are the required differential equations describing tran-
sient particle motion during passage of a plane wave. The constants C p and C s ap-
pearing in the equations are wave propagation velocities. In a manner analogous to
the bar problem, the general solutions of the wave equations are found to be
u x = f 1 (x − C p t) + F 1 (x + C p t) (10.36)
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