Combustion, Explosion, and Shock Waves, Vol. 39, No. 1, pp. 115 118, 2003
Research into the Effect of Loosening in Failed Rock
Yu. S. Vakhrameyev1 UDC 539.379
Translated from Fizika Goreniya i Vzryva, Vol. 39, No. 1, pp. 132 136, January February, 2003.
Original article submitted January 21, 2002.
The loosening phenomenon is examined by solving simplified problems about con-
verging and diverging motions of spherical layers, and also by an experimental study
of a plane motion of a substance undergoing shear deformations. An expression for
the loosening function is obtained; this expression predicts variation of density under
shear deformations at fixed pressure in the range 1.67 < Á < 2.3 g/cm3.
Key words: loosening, internal friction, shear deformation, data processing, exca-
vating explosion.
1. In analyzing results of large-scale excavating (all soils are divided into four groups according to their
explosions previously performed in the USA and USSR, clay content and fragment roundness, i.e., parameters
mechanical strength was excluded from the set of pa- affecting internal friction).
rameters specifying the properties of the medium; at 2. In the absence of mechanical strength, the edge
the same time, internal (dry) friction and rock-loosening of the true crater is the border between the region where
characteristics were adopted as the most important pa- the soil moves and the region where the soil is  blocked
rameters affecting the cratering phenomenon. This ap- owing to violated fluidity conditions. It is seen from here
proach substantially facilitates classification of excavat- how important proper account of the resisting force to
ing results, since it is normally impossible to reliably motion acting on the failed substance is.
measure the mechanical strength of a medium because In a loosable, cohesionless medium, the shear
of voids and cracks present in it. For the same reason, strength is related not only to friction but also to over-
the effective mechanical strength of any soil is much coming of pressure forces acting to prevent changes in
lower than the mechanical strength of its individual volume of the substance. With allowance for loosening,
fragments. the dry-friction coefficient k in the well-known Prandtl
Thus, the medium in which an explosion occurred Reuss relation should be replaced by the sum k + Åš,
was considered as a loose substance consisting of densely where Åš is a dimensionless function (loosening func-
packed fragments. To describe the substance, the tion) that enters the equation predicting the variation
present author advanced a failed-medium model [1]. of material density Á under a constant-pressure shear:
The adopted approach was validated by the fact that,
d ln Á
= -Åš(p, Á). (2)
within this concept, we were able to successfully classify
dł
results of almost all large-scale explosions and, based on
Here Å‚ is the effective shear angle (in radians) and
the results withdrawn from the tests of [2, 3], describe
all the maximal crater-volume data by a single formula: dł = 2/3 J2dt
1/3
Vmax = C(E0/Á0)1/3.6. (1)
= 2/3 (e1 - e2)2 + (e2 - e3)2 + (e3 - e1)2 dt,
Here Á0 is the soil density and E0 is the explosion power.
where J2 is the second invariant of the strain-rate ten-
The effective explosion power for nuclear charges is as-
sor deviator, and ei are the principal components of the
sumed to be lower than that of charges prepared from
strain-rate tensor. A physical, or, more exactly, geo-
ordinary explosives by a factor of 1.35. The value of the
metric interpretation of formula (2) is given in [2]. The
parameter C depends on the type of soil under study
Prandtl Reuss relation can easily be refined by sum-
1
ming the heating energy dQ = pk(V dł) in the volume
Å»
Zababakhin Institute of Technical Physics, Russian Federal
Nuclear Center, Snezhinsk 456770; [e-mail]. V and the work dW = pdV = p(-dÁ/Á)V = pÅš(V dÅ‚).
Å» Å» Å»
0010-5082/03/3901-0115 $25.00 © 2003 Plenum Publishing Corporation 115
116 Vakhrameyev
Fig. 1. The  Wedge setup for shearing tests (the front panel is not shown).
Certainly, loosening should also be taken into ac- a section between them, inclined to the horizon at an
count to estimate the actual volume of the failed soil. angle Ä…. The substance experiences shear deformation
3. In spite of the important role of loosening, this two times: as it enters and as it leaves the inclined
phenomenon in failed substances with densely packed section. The total shear is Å‚ = 2Ä…. The deformation
fragments was not adequately covered by previous stud- of the substance occurs under the action of its weight,
ies. with no participation of any other external force. The
Prior to practical tests, the effect of soil loosening pressure in the substance is H"1 kPa.
on the motion of the medium was studied by the present In the  Levers setup (Fig. 2), the specimen is
author by solving analytical problems in simple formu- placed into a parallelepiped where the inclination an-
lations. The motion of a spherosymmetrical layer of gle of two parallel sidewalls can be controllably varied.
an incompressible (but capable of loosening) failed sub- The substance is sheared by an external force applied
stance with dry friction was considered. The following to these walls. In this case, typical pressures are two or-
relations between the radial (pr) and angular (pÕ = p¸) ders of magnitude higher than in tests on the  Wedge
pressure components were obtained: setup.
"
Three substances were tested: failed granite, failed
pÕ 3 Ä… (k + Åš)
= " . (3)
siltstone (hardened clay sand mixture), and granular
pr 3 " 2(k + Åš)
alluvium. The initial porosity of the specimens was
The upper and lower signs refer to the motion to-
changed by compacting. On both setups, the upper
ward and away from the center of symmetry (the lat-
surface of specimens was open.
ter problem was solved by the author in cooperation
Before the tests, the dimensionless function Åš(p, Á)
with Dem yanovsky [4]). It follows from (3) that the
was assumed to be a function of one dimensionless pa-
centrifugal motion is always possible, whereas cen-
"the
rameter Ámin(p)/Á, where Ámin(p) is the equation of
tripetal motion is possible only if (k + Åš) < 3/2. As
"
the limiting load curve on the plane (p, Á) for a given
(k+Åš) 3/2, the pressure component pÕ tends to in-
failed substance (at p = 0, Ámin is the bulk den-
finity, causing  blockage of the substance. Solving the
sity of the substance). The tests performed on the
problem for a cylindrical layer reveals a similar pattern.
 Wedge setup revealed a more complicated pattern:
Loosening can be best examined if the flow is not
at identical Ámin/Á ratios, siltstone and alluvium dis-
interrupted by  blockage. Unfortunately, such flows
played lower loosening than granite. In this connec-
are hard to be organized in laboratory tests, and in
tion, the present author put forward an assumption
practice one has to be satisfied with a plane motion
that natural moisture could induce impurity-related co-
whose properties are intermediate between the outward
hesion between rock fragments. Later, this assumption
and inward spherical motions. As a result,  blockage
was verified by tests performed on the  Levers setup,
of the substance seems to be possible here, which proved
where compression forces in the substance were much
to be the case in actual tests.
stronger than cohesion forces. Here, all the substances
Loosening tests2 were conducted on two setups.
displayed identical dependences of loosening on the pa-
On the  Wedge setup (Fig. 1), the specimen slides
rameter Ámin(p)/Á.
over a surface formed by two horizontal sections and
On the  Levers setup, other deficiencies were
2
The experiments were carried out by staff members of
found: the useful signal was distorted by disturbances
the Zababakhin Institute of Technical Physics of the Rus-
induced by the  blockage zones in the specimens. The
sian Federal Nuclear Center A. F. Vasil ev, A. A. Shakhov,
disturbances are weaker if the shape of the specimen
A. P. Ivaneev, S. N. Kosorukov, A. M. Zasypkin,
V. N. Tolochek, et al. The final data treatment was per- is changed from a skew to rectangular parallelepiped
formed by the present author.
Research into the Effect of Loosening in Failed Rock 117
Fig. 2. The  Levers setup for shearing tests.
and not in the opposite direction. However, even in
this case, at the initial wall inclination angle ² = 30ć%,
disturbances arise already as the inclination angle in-
creases by 10 15ć%. A better insight into the observed
phenomenon was furnished by numerical simulations by
P. Yu. Tverdokhlebov.
In spite of all difficulties, we succeeded in construct-
ing the sought function Åš(Ámin/Á). It was based on data
for granite obtained on the  Wedge setup and data
obtained for small shear angles on the  Levers setup.
The initial density of granite specimens was chosen to
be close to 2.3 g/cm3 (Ámin 1.67 g/cm3).
The tests showed that the effect of loosening de-
pended on the size distribution of rock fragments. For
this reason, the granulometric composition of rocks was
Fig. 3. Relative change in the specific volume versus
chosen such that to be close, on the average, to the nat-
the effective shear angle of granite specimens: radii
ural one characteristic of crushed rocks excavated by
of the circles correspond to 10% deviation from the
real explosions.
weighted average values of "v/v0; the dashed curve
In spite of the considerable scatter of some part
shows the dependence ("v/v0)max = const (limiting
of the data obtained, the spread of averaged values of value).
test parameters, including the granulometric composi-
tion, turned out to be rather small. The averaging was
of rock densities 1.67 < Á 2.3 g/cm3, we believe that
performed based on data gained in 5 6 tests.
the constructed dependence may also be valid at higher
The gained points on the ("v/v0, Å‚) plane can be
densities (for instance, up to Á = 2.4 g/cm3).
successfully fitted with a single curve (Fig. 3). Based
The tests were carried out at low pressures. The
on the graph, Zharikov constructed an analytical de-
possibility of extending the results to the case of high
pendence Åš(Ámin/Á) of the form
pressures is not proved. However, this method is self-
Åš = Z(1.5 + 0.769Z - 38.8Z2 + 218Z3),
consistent and satisfies the limiting cases. At high pres-
sures, the density Ámin increases abruptly, and the effect
Z = 1 - Ámin(p)/Á.
due to loosening appreciably diminishes, so that no spe-
In Fig. 3, v0 = 1/Á0 is the specific volume of the sub- cial precision in its description is necessary.
stance prior to shearing tests, v is the specific volume Since the effects of loosening coincide for all the
after the tests, and "v = v - v0. Although the above three materials with different friction coefficients and
expression was obtained by treating data in the range fragment roundness, the gained function shows much
118 Vakhrameyev
promise in predicting processes in other failed materi- Isotropy of material properties is an assumption of
als. The adopted model also allows one to predict the fundamental importance for the present model. In re-
density of the substance at the landing moment of freely ality, this assumption can be violated. Anisotropy is
dispersed fragments. The latter follows from geometric hard to be taken into account; however, its influence is
considerations. Here, the friction coefficient plays no manifested, especially in loose substances. It was noted
part at all. that, in a medium that experienced one-sided loading,
4. Finally, let us dwell on other two points con- the behavior of the substance under subsequent defor-
cerning the effect of loosening. mations depends on the strain direction. Anisotropy
4.1. Although the effect of loosening has no direct seems to be the reason for the above-noted considerable
influence on the amount of released heat, the shear- spread of measurement results in shearing tests, espe-
induced dissipation of energy is higher in a loosening cially if the porosity of specimens is high (see Fig. 3).
medium than in a nonloosening one for identical val- Another consequence of anisotropy is the fact that, in
ues of the friction coefficient k. Indeed, since loosen- some reported shearing tests, an increase in sand den-
ing causes an increase of shear-resistance forces, the sity instead of its decrease was observed.
pressure p should necessarily rise for the motion in the Papers [1 4] can be found in: Yu. S. Vakhrameyev,
medium to be preserved, and hence, the energy spent Selected Issues in Explosion and Cumulation Physics
on heating, which is proportional to kp (p is the mean (collected scientific papers) [in Russian], Zababakhin
Å»
value of the principal values of the negative-stress ten- Institute of Technical Physics, Russian Federal Nuclear
sor), also increases. Center, Snezhinsk (1997).
Certainly, all the aforesaid is valid if the increase This work was supported by the International Sci-
in p does not result in motion  blockage . ence and Technology Center (Grant No. 1124 99).
4.2. In Russia, Nikolaevskii was the first to draw
attention of researchers to the dilatancy phenomenon;
the same author gave a short review of foreign publica-
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Atom. Nauki Tekh., Ser. Teor. Prikl. Fiz., No. 1, 22 31
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(1994).
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Ser. Teor. Prikl. Fiz., No. 1, 63 72 (1988).
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