Page 307 - Rock Mechanics For Underground Mining

GENERAL DETERMINATION OF RELEASED AND EXCESS ENERGY

Inspection of equations 10.67 and 10.68 shows that, at a relatively long elapsed time
after excavation of the opening, the exponential terms vanish, and the static elastic
solution is recovered. Equation 10.67 also indicates that, for r = a, rr is identically
zero, demonstrating that the boundary condition at the surface of the spherical opening
is satisfied by the solution. It is also noted that, in each of equations 10.67 and 10.68,
the first term on the right-hand side corresponds to the dynamic stress, and the second
term to the static stress.
Insight into the magnitudes of the dynamic stresses and their temporal and local
variations can be obtained directly from equations 10.67 and 10.68. The parameter
T , which is the local reference time for a point in the medium, is defined by

T = t − [(r − a)/C p ] = (a/C p )[(C p t/a) − (r/a) + 1]

Therefore the parameters 0 T and 0 T in equation 10.68 become
0 T = [(1 − 2 )/(1 − )][(C p /a) − (r/a) + 1]

0 T = [(1 − 2 ) 1/2 /(1 − )][(C p t/a) − (r/a) + 1]

The case a = 1m, = 0.25, has been used to determine the temporal variation
of the circumferential boundary stress, and the radial variation of the radial and
circumferential stresses, at various elapsed times after the instantaneous generation
of the spherical cavity. The temporal variation of the boundary stress ratio, shown
Figure 10.11 Temporal variation of graphically in Figure 10.11, indicates that

/p decreases from its ambient value of
boundary stress around a sphere sud- unity immediately after creating the opening. The boundary stress ratio then increases
denly excavated in a hydrostatic stress rapidly to a maximum value of 1.72, at a scaled elapsed time which corresponds to
field.
the maximum radially inward displacement of the cavity surface. The boundary stress
ratio subsequently relaxes, in a manner resembling an over-damped elastic vibration,
to achieve the static value of 1.50. The transient over-stress at the boundary, which
is about 15% of the final static value, is not insignificant. It is also observed that
transient effects at the excavation boundary are effectively completed at a scaled
−1
time of about 8, corresponding, for C p = 5000 ms , to a real elapsed time of about
1.6 ms.
The radial variations of the radial-and circumferential stress ratios, shown in
Figure 10.12, confirm that the excavation process initiates a stress wave at the cavity
surface. This radiates through the medium at the longitudinal wave velocity, before
subsequent achievement of the static radial and circumferential stress distributions
around the opening. This general view, that the excess energy mobilised locally by the
sudden reduction of the surface forces, must be propagated to the far field to establish
localequilibrium,isentirelycompatiblewithearlierconsiderationsofmining-induced
energy changes.

10.5 General determination of released and excess energy

In later discussion, it is shown that empirical relations can be established between
released energy and the occurrence of crushing and instability around excavations.
The preceding discussion indicated the relation between transient under-stressing
and overstressing of the medium surrounding a suddenly developed excavation, and

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