Research into the Effect of Loosening in Failed Rock, CHEMIA I PIROTECHNIKA, Chemia i Pirotechnika

[ Pobierz całość w formacie PDF ]
Combustion, Explosion, and Shock Waves, Vol. 39, No. 1, pp. 115{118, 2003
Research into the Eect of Loosening in Failed Rock
Yu. S. Vakhrameyev
1
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 simplied 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 xed pressure in the range 1:67 < < 2:3 g/cm
3
.
Key words: loosening, internal friction, shear deformation, data processing, exca-
vating explosion.
1. In analyzing results of large-scale excavating
explosions previously performed in the USA and USSR,
mechanical strength was excluded from the set of pa-
rameters specifying the properties of the medium; at
the same time, internal (dry) friction and rock-loosening
characteristics were adopted as the most important pa-
rameters aecting the cratering phenomenon. This ap-
proach substantially facilitates classication of excavat-
ing results, since it is normally impossible to reliably
measure the mechanical strength of a medium because
of voids and cracks present in it. For the same reason,
the eective mechanical strength of any soil is much
lower than the mechanical strength of its individual
fragments.
Thus, the medium in which an explosion occurred
was considered as a loose substance consisting of densely
packed fragments. To describe the substance, the
present author advanced a failed-medium model [1].
The adopted approach was validated by the fact that,
within this concept, we were able to successfully classify
results of almost all large-scale explosions and, based on
the results withdrawn from the tests of [2, 3], describe
all the maximal crater-volume data by a single formula:
(all soils are divided into four groups according to their
clay content and fragment roundness, i.e., parameters
aecting internal friction).
2. In the absence of mechanical strength, the edge
of the true crater is the border between the region where
the soil moves and the region where the soil is \blocked"
owing to violated uidity conditions. It is seen from here
how important proper account of the resisting force to
motion acting on the failed substance is.
In a loosable, cohesionless medium, the shear
strength is related not only to friction but also to over-
coming of pressure forces acting to prevent changes in
volume of the substance. With allowance for loosening,
the dry-friction coecient k in the well-known Prandtl{
Reuss relation should be replaced by the sum k + ,
where is a dimensionless function (loosening func-
tion) that enters the equation predicting the variation
of material density under a constant-pressure shear:
d ln
d
= (p;):
(2)
Here is the eective shear angle (in radians) and
q
2=3
d =
J
0
2
dt
max
= C(E
0
=
0
)
1=3:6
:
dt;
where J
0
2
is the second invariant of the strain-rate ten-
sor deviator, and e
i
are the principal components of the
strain-rate tensor. A physical, or, more exactly, geo-
metric interpretation of formula (2) is given in [2]. The
Prandtl{Reuss relation can easily be rened by sum-
ming the heating energy dQ = pk(V d) in the volume
V and the work dW = pdV = p(d=)V = p(V d).
0010-5082/03/3901-0115 $25.00
c
2003
Plenum Publishing Corporation
(1)
=
q
2=3
(e
1
e
2
)
2
+ (e
2
e
3
)
2
+ (e
3
e
1
)
2
Here
0
is the soil density and E
0
is the explosion power.
The eective explosion power for nuclear charges is as-
sumed to be lower than that of charges prepared from
ordinary explosives by a factor of 1.35. The value of the
parameter C depends on the type of soil under study
1
Zababakhin Institute of Technical Physics, Russian Federal
Nuclear Center, Snezhinsk 456770; nto2@vniitf.ru.
115
V
1=3
116
Vakhrameyev
Fig. 1. The \Wedge" setup for shearing tests (the front panel is not shown).
Certainly, loosening should also be taken into ac-
count to estimate the actual volume of the failed soil.
3. In spite of the important role of loosening, this
phenomenon in failed substances with densely packed
fragments was not adequately covered by previous stud-
ies.
a section between them, inclined to the horizon at an
angle . The substance experiences shear deformation
two times: as it enters and as it leaves the inclined
section. The total shear is = 2. The deformation
of the substance occurs under the action of its weight,
with no participation of any other external force. The
pressure in the substance is 1 kPa.
In the \Levers" setup (Fig. 2), the specimen is
placed into a parallelepiped where the inclination an-
gle of two parallel sidewalls can be controllably varied.
The substance is sheared by an external force applied
to these walls. In this case, typical pressures are two or-
ders of magnitude higher than in tests on the \Wedge"
setup.
Three substances were tested: failed granite, failed
siltstone (hardened clay{sand mixture), and granular
alluvium. The initial porosity of the specimens was
changed by compacting. On both setups, the upper
surface of specimens was open.
Before the tests, the dimensionless function (p;)
was assumed to be a function of one dimensionless pa-
rameter
min
(p)=, where
min
(p) is the equation of
the limiting load curve on the plane (p;) for a given
failed substance (at p = 0,
min
is the bulk den-
sity of the substance). The tests performed on the
\Wedge" setup revealed a more complicated pattern:
at identical
min
= ratios, siltstone and alluvium dis-
played lower loosening than granite. In this connec-
tion, the present author put forward an assumption
that natural moisture could induce impurity-related co-
hesion between rock fragments. Later, this assumption
was veried by tests performed on the \Levers" setup,
where compression forces in the substance were much
stronger than cohesion forces. Here, all the substances
displayed identical dependences of loosening on the pa-
rameter
min
(p)=.
On the \Levers" setup, other deciencies were
found: the useful signal was distorted by disturbances
induced by the \blockage" zones in the specimens. The
disturbances are weaker if the shape of the specimen
is changed from a skew to rectangular parallelepiped
Prior to practical tests, the eect of soil loosening
on the motion of the medium was studied by the present
author by solving analytical problems in simple formu-
lations. The motion of a spherosymmetrical layer of
an incompressible (but capable of loosening) failed sub-
stance with dry friction was considered. The following
relations between the radial (p
r
) and angular (p
'
= p
)
pressure components wer
e
obtained:
p
'
p
r
=
p
3 (k + )
p
3 2(k + )
:
(3)
The upper and lower signs refer to the motion to-
ward and away from the center of symmetry (the lat-
ter problem was solved by the author in cooperation
with Dem'yanovsky [4]). It follows from (3) that the
centrifugal motion is always possible, whereas the cen-
(k+) !
p
3=2, the pressure component p
'
tends to in-
nity, causing \blockage" of the substance. Solving the
problem for a cylindrical layer reveals a similar pattern.
Loosening can be best examined if the ow is not
interrupted by \blockage." Unfortunately, such ows
are hard to be organized in laboratory tests, and in
practice one has to be satised with a plane motion
whose properties are intermediate between the outward
and inward spherical motions. As a result, \blockage"
of the substance seems to be possible here, which proved
to be the case in actual tests.
Loosening tests
2
were conducted on two setups.
On the \Wedge" setup (Fig. 1), the specimen slides
over a surface formed by two horizontal sections and
2
The experiments were carried out by sta members of
the Zababakhin Institute of Technical Physics of the Rus-
sian Federal Nuclear Center A. F. Vasil'ev, A. A. Shakhov,
A. P. Ivaneev, S. N. Kosorukov, A. M. Zasypkin,
V. N. Tolochek, et al. The nal data treatment was per-
formed by the present author.
tripetal motion is possible only if (k + ) <
p
3=2. As
 Research into the Eect 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 = 30
,
disturbances arise already as the inclination angle in-
creases by 10{15
. A better insight into the observed
phenomenon was furnished by numerical simulations by
P. Yu. Tverdokhlebov.
In spite of all diculties, we succeeded in construct-
ing the sought function (
min
=). 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/cm
3
(
min
' 1:67 g/cm
3
).
The tests showed that the eect of loosening de-
pended on the size distribution of rock fragments. For
this reason, the granulometric composition of rocks was
chosen such that to be close, on the average, to the nat-
ural one characteristic of crushed rocks excavated by
real explosions.
In spite of the considerable scatter of some part
of the data obtained, the spread of averaged values of
test parameters, including the granulometric composi-
tion, turned out to be rather small. The averaging was
performed based on data gained in 5{6 tests.
The gained points on the (v=v
0
;) plane can be
successfully tted with a single curve (Fig. 3). Based
on the graph, Zharikov constructed an analytical de-
pendence (
min
=) of the form
= Z(1:5 + 0:769Z 38:8Z
2
+ 218Z
3
);
Fig. 3. Relative change in the specic volume versus
the eective shear angle of granite specimens: radii
of the circles correspond to 10% deviation from the
weighted average values of v=v
0
; the dashed curve
shows the dependence (v=v
0
)
max
= const (limiting
value).
of rock densities 1:67 < 6 2:3 g/cm
3
, we believe that
the constructed dependence may also be valid at higher
densities (for instance, up to = 2:4 g/cm
3
).
The tests were carried out at low pressures. The
possibility of extending the results to the case of high
pressures is not proved. However, this method is self-
consistent and satises the limiting cases. At high pres-
sures, the density
min
increases abruptly, and the eect
due to loosening appreciably diminishes, so that no spe-
cial precision in its description is necessary.
Since the eects of loosening coincide for all the
three materials with dierent friction coecients and
fragment roundness, the gained function shows much
Z = 1
min
(p)=:
In Fig. 3, v
0
= 1=
0
is the specic volume of the sub-
stance prior to shearing tests, v is the specic volume
after the tests, and v = v v
0
. Although the above
expression was obtained by treating data in the range
118
Vakhrameyev
promise in predicting processes in other failed materi-
als. The adopted model also allows one to predict the
density of the substance at the landing moment of freely
dispersed fragments. The latter follows from geometric
considerations. Here, the friction coecient plays no
part at all.
4. Finally, let us dwell on other two points con-
cerning the eect of loosening.
4.1. Although the eect of loosening has no direct
inuence on the amount of released heat, the shear-
induced dissipation of energy is higher in a loosening
medium than in a nonloosening one for identical val-
ues of the friction coecient k. Indeed, since loosen-
ing causes an increase of shear-resistance forces, the
pressure p should necessarily rise for the motion in the
medium to be preserved, and hence, the energy spent
on heating, which is proportional to kp (p is the mean
value of the principal values of the negative-stress ten-
sor), also increases.
Certainly, all the aforesaid is valid if the increase
in p does not result in motion \blockage".
4.2. In Russia, Nikolaevskii was the rst to draw
attention of researchers to the dilatancy phenomenon;
the same author gave a short review of foreign publica-
tions on the behavior of soft materials under shearing
[5]. It was believed that changes in density might be
both positive and negative. In view of this, the phe-
nomenon was called \dilatancy" and not loosening.
There is no doubt that dense materials loosen up
under shear at xed pressure. For this reason, compact-
ing, if any, can most likely be observed in low-density
media.
As a matter of fact, this cannot be the case in an
isotropic medium. Indeed, had the function (
min
=)
be changing its sign at some value of =
cr
, so that
the density be increasing under shearing, then in some
cases, the shear would lead to loosening and in other
cases to compacting at identical characteristics of the
material (density, granulometric composition, and pres-
sure). This is possible only if the properties of the
medium depend not only on the above-listed parame-
ters but also on other parameters ignored by the present
model.
Isotropy of material properties is an assumption of
fundamental importance for the present model. In re-
ality, this assumption can be violated. Anisotropy is
hard to be taken into account; however, its inuence is
manifested, especially in loose substances. It was noted
that, in a medium that experienced one-sided loading,
the behavior of the substance under subsequent defor-
mations depends on the strain direction. Anisotropy
seems to be the reason for the above-noted considerable
spread of measurement results in shearing tests, espe-
cially if the porosity of specimens is high (see Fig. 3).
Another consequence of anisotropy is the fact that, in
some reported shearing tests, an increase in sand den-
sity instead of its decrease was observed.
Papers [1{4] can be found in: Yu. S. Vakhrameyev,
Selected Issues in Explosion and Cumulation Physics
(collected scientic papers) [in Russian], Zababakhin
Institute of Technical Physics, Russian Federal Nuclear
Center, Snezhinsk (1997).
This work was supported by the International Sci-
ence and Technology Center (Grant No. 1124{99).
REFERENCES
1. Yu. S. Vakhrameyev, \Model of a failed medium," Vopr.
Atom. Nauki Tekh., Ser. Teor. Prikl. Fiz., No. 1, 22{31
(1994).
2. Yu. S. Vakhrameyev and N. G. Mikhal'kov, \Similarity
between underground explosions and approximate mod-
eling of excavating explosions," Vopr. Atom. Nauki Tekh.,
Ser. Teor. Prikl. Fiz., No. 1, 63{72 (1988).
3. Yu. S. Vakhrameyev, \Physical foundations for approxi-
mately modeling explosions with ejecta," Combust. Expl.
Shock Waves, 31, No. 1, 120{125 (1995).
4. Yu. S. Vakhrameyev and S. V. Dem'yanovskii, \Void ex-
pansion in a loosening medium with dry friction," Fiz.
Tekh. Probl. Razrab. Polezn. Iskop., No. 1, 38{42 (1974).
5. V. N. Nikolaevskii, \Relation between bulk and shear de-
formations and shock waves in loose soils," Dokl. Akad.
Nauk SSSR, 177, No. 3, 542{545 (1967).
[ Pobierz całość w formacie PDF ]

  • zanotowane.pl
  • doc.pisz.pl
  • pdf.pisz.pl
  • tejsza.htw.pl

    •