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35
PROBLEM 4
EXPLOSIVELY INDUCED STRESS WAVE FROM A FINITE CYLINDRICAL SOURCE
1. INTRODUCTION
Blasting is a fundamental technique in mining operations, and there is a
long history of attempts to analyse the sequence of events in the vicinity of
an explosion in rock. At present factors such as amount of explosive, place-
ment and depth of holes, type of detonator and speed of explosive, are decided
on experi ence, and coul d proba bly be improved, wi th great savings to the
industry, if there were a more analytic basis available.
The problem of "subsonic" radiation of elastic waves is well within the
domain of classical applied mathematics, and several papers have been pub-
lished. A little less conventional is the analysis of radiation from a deto-
nation wave travelling faster than the elastic waves, but there is at least
one adequate paper in this area (Paplinski, 1977), and the acousti c analogy
works quite well.
All this analysis fails if the medium begins to fail as the elastic waves
pass through, affecti ng thei r propagati on. A major probl em is to deci de
whether the propagation of cracking, probably by hot gases, is in some region
faster than the elastic waves; or whether the elastic waves, probably modi-
fied by reflection, precede and cause the craCking.
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2. CLASSICAL ANALYSIS
A well-known early paper in the field is that of Heelan (1953). Like
several that followed, it considers a pressure instantaneously applied to a
portion of the wall of an infinite cylinder in an infinite medi un , Advanced
and elegant mathematic methods produce asymptotic wave forms of a relatively
easily calculated shape. Unfortunately, as asserted by Jordan (1962) and
demonstrated more convincingly by Abo-Zeno (1977), the results are wrong.
Jordan does convincingly shov , by trying to do the analysis correctly, that no
easy answers are to be obtained that way.
The wave propagation problem treated by those three authors is now
described. Two elastic potentials 4> and 1jJ are defined which determine the
radial and axial displacements ur and Uz by the equations
u = 04> _ 01jJ
r or oz'
=~+~~
Uz oz r or
The potentials satisfy the wave equations
~/p, in which A,~ are the Lame constants.
The boundary condition is an impulsive rise in the radial stress
Heelan allows non-radial symmetry. An important difficulty wi th these methods
is the reliance on the maintenance of an infinite cylindrical boundary. It is
possible to impose various pressure conditions there, but both the finiteness
of the cylinder and the modification by the explosion of parts of the wall are
37
difficul t to include.
Jordan solves the p.d.e. by taking Fourier transforms with respect to t
and z , leaving a 8essel equation in r. The result gives transformed stresses
as linear functions of HO(I;R), H,(r,;R), Ho(CR), H,(CR) where R is the radial
co-ordinate, C and r,; are given by
2 2o + (s/CT) ,
i + (s/CL)2,
sand 0 are the transform variables, and the denominator includes a term
22r,; s 2 2a c2 H, (al;)/H, (aC) + (0 +C )H, (aC)/Ho(ar,;)
T
2- 4 0 C r,; H,(ar,;)Ho(ac)
in which a is the radius of the hole. Jordan then replaces the Bessel fune-
tions by large argument asymptotic expansions to show the emergence of a dila-
tion wave front and a shear wave front travelling wi th different speedS (pre-
viously shown by Selberg '952).
Abo-Zena ('977) covers much the same ground to produce the response to a
6-function in the radial stress
The displacements are found to be
4:~L [, - '. 'I [ R Zo 1u A+~ cos r,; F{tO'zO) 6 t - CL - :: · CL] cost; ,r
2
• • [ Rau 2RC" cos z slnl; F(tO'zO) 0 t - C
T
- to + C
T
cosr,; .z T
3. SHOCKWAVE SOLUTIONS
Shocks were briefly treated in Jordan's paper, and more extensively by
Paplinski and Wlodarczyk ('977). We decided treatment of this case was essen-
tial because commonly quoted detonation speeds of 4-'0 km/sec. exceeded common
38
P-wave velocities of 2-3.5 km/sec. The details of Paplinski's analysis are
reminiscent of that of Jordan, but are scmewhat simpler. The detonation wave
is given uniform velocity V, leading to a linear function for the radial dis-
placement velocity
VrCf;,n) = ~ [_1_ cSCrrf; ) + HCrrf;!l) f f Cf;,p)e-Pn dP]
M J1"!l 0 v
where
and f;, nand f;B are defined by
!lCe-n ,
4. SOLUTION BY SUPERPOSITION OF SPHERICAL SOLUTIONS
In a 1969 paper, Favreau developed a solution with spherical symmetry
which used the gas equations of state to provide a boundary condition allowing
for the effect of expansion of the cavity on the gas pressure. The result
gi ves travelling wave solutions which can be determined frcm a second order
ordinary linear differential equation with constant coefficients. This would
also have been true had no allowance been made for gas expansion CSelberg
1953), the only difference being in the complexity of the coefficients.
Although the mathematics is rendered only slightly more complicated,
there is also little improvement in physical realism, because rock is so much
stiffer a material than gas. A 1% increase in cavity radius diminishes the
gas pressure by scmething like 3%, but the change in rock stress is enormous.
The notes of Bland C1980) reinforce thi s point.
One can try to incorporate the same effect in a blast moving up a
cylindrical borehole by superimposing spherical explosions translated in time
39
and space. But there are several drawbacks. Favreau's analysis based on gas
confined in a sphere is now applied to gas not so confined. Energy moving up
the borehole in the gas phase and as surface Rayleigh waves is not allowed
for. And in common with most solutions, no account is taken of the crumbling
region behind the detonation front.
5. CONCLUSIONS
The method of superposition of spherical solutions of the type due to
Favreau should not be used because the assumption of radial gas expansion is
not justified, and Favreau's refinement of Selberg's notation is neither sig-
nificant nor applicable in this case. The most attractive replacement is a
conical wave solution, probably a shock wave. This idea is also suggested by
Bland, and some of the theory in Section 3 should be useful.
REFERENCES
Selberg, H.L., "Transient compression waves from spherical and cylindrical
cavi ti es", Ark. Fys. 5 (1952), 97-108.
Heel an , D.A., "Radiation from a cylindrical source of finite length", Geophy-
sics 18 (1953),185-696.
Jordan, D.W., "The stress wave from a finite cylindrical explosive source", J.
Math. Mech. 11 (1962), 503-551.
Favreau, R.F. "Generation of strain waves in rock by an explosion in a spheri-
cal cavity", J. Geophys Res. 74 (1969), 4267-4280.
Abo-Zena, A.M. "Radiation from a finite cylindrical explosive source", Geophy-
sics 42 (1977), 1384-1393.


Explosively induced stress wave from a finite cylindrical source

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