Page 303 - Rock Mechanics For Underground Mining

SPHERICAL CAVITY IN A HYDROSTATIC STRESS FIELD

This implies that, for the cylindrically divergent wave

u r ∝ 1/r 1/2 (10.46)

10.4 Spherical cavity in a hydrostatic stress field

The purpose in this section is to examine the relative magnitudes of the static and
dynamic stresses associated with creating an excavation, to illustrate methods of
evaluating released and excess energy, and to correlate the dynamic stresses with
excess energy magnitude. The reason for choosing a spherical opening for study
is that the problem is analytically tractable. A two-dimensional problem, such as a
cylindrical excavation, cannot be treated productively due to the lack of a closed form
solution for the response to a step increase in pressure to the internal surface of the
hole. The problem of the spherical opening has been considered by Hopkins (1960),
Timoshenko and Goodier (1970), and Bray (1979), on whose work the following
discussion is based.
Figure 10.10 shows a diametral section of a spherical opening, of radius a,ina
medium subject to a hydrostatic far-field stress, of magnitude p. Relative to spherical
polar (r,
, ) co-ordinate axes, total stresses after excavation and excavation-induced
displacement are given, according to Poulos and Davis (1974), by

3
3
rr = p[1 − (a /r )]
3
3

= = p[1 + (a /2r )] (10.47)
r
= r =
= 0
3
u r =−pa /4Gr 2 (10.48)
u
= u = 0

Figure 10.10 Diametral section
through a sphere in a medium subject
to hydrostatic stress.

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