Page 305 - Rock Mechanics For Underground Mining

SPHERICAL CAVITY IN A HYDROSTATIC STRESS FIELD

is independent of the method of excavating the cavity. Therefore the excess energy
W e associated with the spherical stress wave is obtained from equations 10.51 and
10.52, i.e.
2 3
2 3
2 3
W e = ( p a /G) − ( p a /2G) = p a /2G (10.54)
Equation 10.54 defining the excess energy, is identical to equation 10.53 defining
the released energy. W e is therefore confirmed to be the energy imbalance which
arises from the impulsive unloading of a rock internal surface to form a traction free
excavation surface.
In order to relate the excess energy to the magnitudes of the dynamic stresses
induced by sudden excavation, it is necessary to consider details of wave propagation
in an elastic medium. Corresponding to the differential equations of equilibrium for
a static problem are the differential equations of motion for a dynamic problem. For
the spherically symmetric problem, the equation of motion (equation 10.40) may be
written as
2
2
2
2
2

∂ u r /∂r + (2/r)∂u r /∂r − 2u r /r = 1/C ∂ u r /∂t 2 (10.55)
p
where C p is the longitudinal wave velocity.
For the diverging wave, i.e. propagating radially outwards, the general solution to
equation 10.55 is of the form
2

u r = (1/r) f (r − C p t) − (1/r ) f (r − C p t) (10.56)
where the nature of the function f is chosen to satisfy the initial and boundary con-
ditions for a particular problem. Sharpe (1942) established a solution for equation
10.56 for the case of a varying pressure p(t) applied to the surface of a spher-
ical cavity, of radius a. For an exponentially decaying applied internal pressure,
given by
p(t) = p 0 exp(− t)

Sharpe found that the function f satisfying equation 10.56 is given by
f = exp(− 0 T )(A cos 0 T + B sin 0 T ) − A exp(− T ) (10.57)

where

0 = (C p /a)[(1 − 2 )/(1 − )]
0 = (C p /a)[(1 − 2 ) 1/2 /(1 − )]
T = t − [(r − a)/C p ]
2 2
A = P 0 a/ + ( 0 − ) (10.58)
0
(10.59)
B = A( 0 − )/ 0
For the case of a spherical opening suddenly developed in a medium subject to
hydrostatic stress p, the induced displacement field can be determined from Sharpe’s
solutionbyputting p 0 =−p and = 0.Thissatisfiestherequiredboundarycondition

287