Page 296 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

Figure 10.6 Problem definition and
elementary body for analysis of a
longitudinal wave in a bar.

the element, it is necessary to introduce an inertial (d’Alembert) force opposing the
sense of motion, given in magnitude by d¨ u x . If this force is introduced, the forces on
the element may be treated as an equilibrating system, i.e.

xx A − dM ¨ u x − [ xx + (∂ xx /∂ x )dx]A = 0 (10.8)

where A is the cross-sectional area of the bar.
Since dM = A dx, where is the material density, and
2 2
∂ xx /∂ x = (∂/∂x)Eε xx =−E ∂ u x /∂x
equation 10.8 becomes

2
2
2
∂ u x /∂t = (E/ )∂ u x /∂x 2 (10.9)
Equation 10.9 is the differential equation for particle motion in the bar, or the bar
wave equation. The general solution of the equation is of the form
u x = f 1 (x − C B t) + f 2 (x + C B t) (10.10)

where f 1 and f 2 are functions whose form is determined by the initial conditions, i.e.
the manner of initiation of the wave. It is readily demonstrated, by differentiation, that
the expression for u x satisfies equation 10.9, provided C B is defined by the expression
1
C B = (E/ ) 2 (10.11)

C B is called the bar velocity, and represents the velocity of propagation of a pertur-
bation along the bar.
In equation 10.10, the term whose argument is (x − C B t) represents a wave propa-
gating in the positive direction of the co-ordinate axis, i.e. a forward progressive wave.
The term with argument (x + C B t) represents a wave propagating in the negative

278