Abstract—The paper deals with the problems on mechanics of material with gas-filled fissure-like discontinuities, considered in terms of coal, rocks and gas outbursts in coal mines.
Keywords: outburst, gas-filled fissure, stress state
INTRODUCTION
Considering the theory of runs and the related findings [1-10], it is important to state that the growth of gas-filled fissures under the gas content expansion is a constitutive factor for outbursts and associated events in coal mines.
Uniformly compressed in situ coal has an abundance of pores, fissures and other fissure-like joints filled with pressurized gas. Mining induces zones of higher stresses in rocks, nets of oriented fissure arise in these zones and, as a consequence, coal in these zones becomes anisotropic, more yielding and lower elastic. The deformation state of coal depends on the stresses; on the other hand, the strain and strength properties of coal affect the stress distribution, which is important for prediction and prevention of outbursts.
The simplest but critical issue related with the origin and evolution of coal and gas outbursts is the determination of stresses in the vicinity of a blasthole made in the initially distressed and gas-bearing coal seam (Fig. 1). Drilling is a commonly used operation to prevent outbursting or to determine coal seam gas pressure [1].
The drilling distresses the coal seam. The normal compression stresses rσ on sites where normals
coincide with the drilled hole radius direction, decrease and stresses θσ on the perpendicular sites grow;
thus different stress zones originate in the coal seam (Fig. 1).
Fig. 1. Drill-hole perpendicular to the coal seam: 1—elastic zone; 2—zone of oriented fissures; 3—plastic (protective) zone; 4—disintegrated coal zone.
28 KOVALENKO et al.
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
The zone where coal is considered isotropic-elastic is far away from a borehole. In the elastic zone, the highly-pressurized gas expands in pores and fissures, and can initiate the growth of peripheral joints, herewith with no growth of radial jointing. This is explained by the fact that the radial stresses
rσ usually fall while the peripheral stresses θσ grow. This area becomes cylindrically anisotropic,
namely, at every point it gets transversally anisotropic relative to the rotation-symmetry axis, which coincides with the drill-hole radius. The plastically deformed-coal zone adjacent to the drill-hole functions as a protective “plug.”
To evaluate precisely radii of the plastic and anisotropic zones, it is necessary to find expressions for stresses and shears in these zones. However, we face the problem that in the transversally-isotropic zone containing an oriented net of fissures, radial elastic modulus varies relative to the radius as the growth rate of fissures in this zone depends on rσ at the fissure location point.
The behavior of a single gas-filled fracture under decreasing external compression is analyzed in details in [2], the research results are reported in brief in the present paper.
1. SINGLE GAS-FILLED FISSURE
Consider a fissure filled with gas under the pressure 0p in an elastic isotropic medium affected by
the uniform stress 0iσ far from the study fissure. Let the coordinate axes 1x 2x 3x match the direction
of the 0iσ tensor components. Place the study fissure onto the plane 1x 2x .
Assume that at the initial moment the gas pressure in the fissure equals the triaxial compression stress: 0
03
02
01 p−=== σσσ , hereinafter the compression stresses are negative.
In response to variation in the external compression, the fissure behavior will depend on the principal stress 3σ , while the effect of 1σ and 2σ on the fissure is minor. In this connection, it is
reasonable to consider that the stress 3σ , denoted as σ , is variable, while 1σ and 2σ remain
constant. The reduced compression σ differs from the gas pressure in the fissure, р, and this is equivalent to application of tension to the fissure surfaces:
σ+= pT . (1)
When Т reaches the critical value determined by the geomaterial cohesion module K, the fissure starts to grow. The term for the growth of a disk-shaped fissure in an elastic-isotropic material is [11]:
γ=T . (2)
Here, the symbol:
2/10 )2( −= RKγ (3)
is introduced, where 0R is the fissure radius.
Denote the fissure volume and gas pressure in it at the moment of its growth as cV and cp ,
respectively. Disregarding the heat exchange between the material and gas, we have:
kkcc VpVp 00= , (4)
where 0p and 0V are the initial gas pressure in the fissure and the fissure volume; k is the gas
adiabatic index.
DEFORMATION OF A COAL SEAM WITH A SYSTEM OF ISOLATED GAS-FILLED FISSURES 29
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
Under tension, the disk-shaped fissure takes an ellipse shape in the central cross-section, so the variation in this stress by ΔТ causes the variation in the fissure volume [11]:
30
2
3)1(16
TRE
vV Δ−=Δ , (5)
where 0E , 0v are the Young modulus and Poisson ratio for the material inside the fissure.
Then, by (2)-(5) we find the external compression stress, at which the fissure growth starts:
k
c RVE
p−
+−= 3
00*
0
11 γγσ , (6)
where )1(16/3 2* vEE −= .
The fissure growth will inevitably cease. Assuming that during the fissure growth the gas behavior obeys the isothermal law (choking) and using (2)-(5), we drive the equation for a new fissure radius cR :
k
cccc VE
Rp
VE
R−
−−
+=
+−
1
0*
30
066
0*
30 11)( γβγβσβγ , (7)
where 2/10 )/( cc RR=β .
To find effective characteristics of a geomaterial with the oriented gas-filled jointing needs solving the problem of a single fissure propagating in an elastic transversal-isotropic geomedium under pressure.
Variations in the disk-shaped fissure volume in the transversal-isotropic medium with an isotropic plane coinciding with the fissure plane can be evaluated from [12]:
)(3
16 30
0)3( σ+=Δ pRIV . (8)
The normal movement of the fissure surfaces obeys the law:
2/100
0)3( )]()[(
4rRRpIw −+= σπ ,
where r is the distance from the fissure center. Alternatively, according to [13]:
2/10 )(2 rRNw n −= β ,
where N is the stress intensity factor. Using the above two relations, we obtain the fissure growth condition:
KRpIn
=+ 2/10
0)3( )2)((
2 σβ . (9)
Here:
])()([2 1033
2011
011
−−= aaanβ ,
2/12013
033
011
1011
0)3( ]})([{)(
21
aaaBaI −= − , (10)
30 KOVALENKO et al.
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
2/12012
2011
2013
033
011
012
011
013
044
011 ]})()][()([{2)(2 aaaaaaaaaaB −−+−+= ,
and 0ija are the elastic constants of the transversely isotropic medium [14].
By analogy with (6), for stresses initiating growth of fissures, we find:
k
nc
IVI
pI
−
+−= γβγβσ
0)3(0
0)3(
00)3( 23
161
2. (11)
2. EFFECTIVE CHARACTERISTICS
The macroscopic strain of a material with displacement discontinuities is [15]:
dYYFVnVn ikkiikik )()(210 ++= εε . (12)
Hereinafter, the zero-indexed magnitudes refer to a geo material; iV is an ith displacement
discontinuity; in is a unit vector of the normal; Y is a set of characteristics of the displacement discontinuity area; dYYF )( is the share of the displacement discontinuity areas in unit volume with
parameters within dY range. Relation (12) is pure geometrical, so the source of iV is of no value.
That is to say, this relation can be helpful to calculating characteristics of a material containing any heterogeneities, provided that these characteristics can be presented as surfaces of displacement discontinuities. Therefore, the problem is reduced to the calculation of iV , namely, components of the
displacement vector of heterogeneous surfaces. However, to find these components, it is required to solve the problem related to a set of the heterogeneities, which is hardly possible in general case.
In view of the above, the researchers [15] evaluated efficient characteristics of a body with numerous cracks by the approach based on the extended self-consistency method. In the case of the high density net of fissures, this method reduces to the differential procedure. This method is efficient to find the effective characteristics of a material with numerous fissures filled with an elastic-linear material. However, if the material inside fissure deforms nonlinearly, then, to calculate iV in (12)
requires solving the problem on a fissure-like heterogeneity in the nonlinear medium. The mathematical problem is reduced to solving the elasticity problem with the stress-dependent elastic moduli.
However, when considering a unidirectional distressing-free process, it is possible unify the elastic and inelastic components if the calculations are made in terms of the initial stresses in rocks, and all the stresses and strains are assessed in agreement to this initial stress state. In other words, the problem is solved in increments. The elastic moduli are evaluated as a ratio of the current-to-in situ stress difference to the current and in situ strain difference. The absence of difference between the dilatants material problem solution (explicitly included inelastic strains) and the medium with the integrally found effective moduli (as described above) is analogous to the absence of difference between the solutions of the nonlinear elastic problem and the distressing-free plastic problem.
Consider small increments of applied stresses. Suggesting that the combination of parameters, Y, does not alter at the additional loading stage, we have for strain increments, from (12):
dYYFVnVn ikkiikik )()(210 ++= δδδεδε . (13)
DEFORMATION OF A COAL SEAM WITH A SYSTEM OF ISOLATED GAS-FILLED FISSURES 31
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
As the behavior of a non-linear material at small stress increments is much similar to the behavior of a linear-elastic material with deformation characteristics depending on the resultant stress state, we can write:
klikli AV σδδ = , (14)
where the tensor components iklA , characterizing only the given heterogeneity, depend on the
resultant final stress state and are assumed constant at each stage of additional loading. Suggesting that klijklij S δσδε 00 = , where ijklS is the yield tensor for a material between heterogeneities, we find
from (13) and (14): klijklij S σδδε *= . (15)
Here:
dYYFAnAnSS ikljjkliijklijkl )()(210* ++= (16)
can be identified as the instant magnitude of the effective yield tensor at the given additional loading stage. Integrating (15) against (16), it is possible to find the wanted ratio between the stress and strain tensors.
As has been stated above, drilling in a rock mass with gas-filled fissures can induce the peripherally oriented jointing at some distance from a new-made hole. Herewith, the medium in this zone gets transversally isotropic with the axis of the rotation symmetry coinciding with a hole radius. Therefore, the effective characteristics of a local rock are needed to evaluate the radius of the oriented fracturing zone.
3. ORIENTED GAS-FILLED JOINTING SYSTEM
Consider an elastic, isotropic material containing a system of oriented gas-filled fissures, homogeneously and isotropically arranged on planes located normally to the axis 3x . The studied
medium is effectively transversally isotropic, the isotropy axis coincides with the axis 3x . This
medium is characterized by five elastic constants and the relationship between the components of the stress and strain tensors is [14]:
33132212111111 σσσε aaa ++= ,
33132211111222 σσσε aaa ++= ,
33332213111333 σσσε aaa ++= , (17)
234423 2/1 σε a= , 134413 2/1 σε a= ,
12121112 )( σε aa −= ,
where ija are the elastic constants.
Let the fissure density is low, namely, fissures do not interact. In this case, the solution to the problem of a single gas-filled fissure in an infinite elastic isotropic material can be helpful to evaluate the effective characteristics which are wanted to estimate iV in (12). Moreover, in the study case with
32 KOVALENKO et al.
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
a cylindrically oriented system of fissures, the combination of parameters, Y, in (12) is reduced to a single parameter, namely, the radius of fissures, a, at 021 == nn and 13 =n .
To define relationships of the values of iV and the applied stress, we address the research results
reported in [15]. The gas content under pressure p in available fissures causes additional alterations of the fissure volume, thereto, it is explicit that coefficient π/23 =c in [15] is replaced by
33331
3 /)(2 σσπ pc += − , while 1c and 2c remain unchanged. Then, in analogy to [12], for the case
with low density of fissures, we have:
011 /1 Ea = , 001312 / Evaa −== ,
+−
+= dRRFRp
Ev
Ea )(
3)1(161 3
0
20
033 σ
σ, (18)
−−
++= dRRFRvE
vE
va )(
)2(3)1(32)1(2 3
00
20
0
044 ,
where 0E and 0v are the Young modulus and Poisson ratio for the material between fissures; )(RF
is the function of the fissure radius distribution. It is evident from (18) that the gas content in the fissures affects only the magnitude of the modulus 33a , while the other moduli ija are not affected.
Express the gas pressure in the fissures in terms of the applied stress. Under assumption that the deformation process is isothermal, we have for the gas within the fissures:
)( 000 VVpVp Δ+= , (19)
where 0p and 0V are initial gas pressure and fissure volume; p is the current gas pressure in a fissure;
VΔ is the variation in the fissure volume in response to the reduced external stress from the initial value 0σ down to the current value σ . From (19) and (5) we find:
2/1
30
*0
2
30*
30* 4
2
1
2
1
+
++
+−=
R
VEp
R
VE
R
VEp σσ , (20)
where )1(16/32
00* VEE −= .
Equations (4), (17), and (20) describe the law of the non-linear-elastic deformation of a material in case of the low-dense gas-filled fissure network in it.
Let the density of fissures is high. We can use the differentiation procedure to determine instantaneous deformation characteristics at a short additional loading stage [15]. It can be based on the solution to the problem of a single heterogeneity occurring in an infinite medium, and its properties are governed by the rest available heterogeneities. In the case with the oriented system of the gas-filled fissures, the medium will respond to relatively small variations in the external compression stresses as an elastic transversally isotropic material with the isotropy axis situated normal to the planes of the fissures. As the gas pressure in the fissures depends on the external compression stress, the values of the instantaneous modulus of the medium are determined by the gained stress values. In [12] the solution from [16] is used to evaluate iV for a “gas-free” fissure in a
the transversally isotropic medium. The gas content in a fissure alters 3V , but expressions for 1V and
2V remain the same, thus:
DEFORMATION OF A COAL SEAM WITH A SYSTEM OF ISOLATED GAS-FILLED FISSURES 33
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
])([3
1630
033
0)(
3kkkkk ppIRV δσσ −+−= . (21)
Here, 3kδ is the Kronecker symbol; 0p , 03kσ are the initial values and р, 3kσ are the current values of
the gas pressure in the fissures and the respective components of the applied stress tensor:
12/1012
011
044
044
011
2/12012
2011
044
00)2(
0)1( )(
21
]})()[({
−
++−=== aaBaaaaaBaIII . (22)
Equations for В, 0)3(I are expressed through (10), and 0
ija are the elastic constants of the material.
Excluding р from (21) by means of (19), we have:
2/1300
233
0330
30033
0330
3003 }4)]({[
21
)]([21
RVppRVpRVV ασσασσα +−+−+−++−= , (23)
where 0)3(0 3/16 I=α .
Differentiating (21) and (23), we obtain:
332/13000
233
0330
300
330330
30030
)(}4)]({[
)(1
316
kkkkRVppRV
pRVRIV δσδ
ασσασσαδ
+−+−−+−+= . (24)
Then, from (14) and (24) we derive the expression for the components of the tensor iklA :
30223113 3
16RIAA == ,
(25)
+−+−−+−+= 2/13
0002
330330
300
330330
30030
)3(333}4)]({[
)(1
316
RVppRVpRV
RIAασσα
σσα.
The rest components of iklA are equal to zero.
Finally, from (15), (16), (17), and (25), at 021 == nn , 13 =n and the low density of the gas-filled
heterogeneities in the transversally isotropic material, we find: 0
1111 aa = , 01212 aa = , 0
1313 aa = ,
dRRFRRVppRV
pRVaa )(
}4)]({[)(
12
32/13
0002
330330
300
330330
30000
3333
+−+−−+−++=
ασσασσαα
, (26)
dRRFRIaa )(3
16 3004444 += .
For fissures of the equal radius 0R at = NdRRF )( , where N is the number of fissures per unit
volume, we obtain from (26):
01111 aa = , 0
1212 aa = , 01313 aa = ,
v}4)]({[
)(1
2 2/130000
233
0330
3000
330330
300000
3333
+−+−−+−
++=RVppRV
pRVaa
ασσασσαα
, (27)
vIaa 004444 3
16+= ,
34 KOVALENKO et al.
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
where 30NRv = .
Differentiation [15] is reduced to the substitution in (26): 33da for ( 03333 aa − ), 44da for ( 0
4444 aa − ),
and )3(I , I for 0)3(I , 0I . This is performed by replacing 0
ija by ija in Eqs. (10) and (22). Therefrom, as
dNdRRF =)( , we derive the system of differential equations to find 33a and 44a :
+−+−−+−+= 2/13
002
330330
30
330330
30
3033
}4)]({[)(
12 RVppRV
pRVRdNda
ασσασσαα
, (28)
344
316
IRdNda = .
Here, )3(3/16 I=α and it is assumed that the form of the function )(NR is known. The initial
conditions are suggested as:
03333 aa = , 0
4444 aa = at 0=N . (29)
If the oriented jointing system is introduced into an initially elastic material:
0033 /1 Ea = , 00
044 /)1(2 Eva += ,
where 0E and 0v are Young modulus and Poisson ratio.
Calculating 33a , 44a from (28) and (29) at different 33σ and integrating the bond between the
increments of the stress and strain tensors from (17), we establish the relationship between the components of the stress and strain tensors.
4. EQUILIBRIUM, CONSTITUTIVE AND COMPATIBILITY EQUATIONS
Returning to the problem of stress distribution in the neighborhood of a cylindrical borehole made in a coal seam (Fig. 1), it has been mentioned above that the dependence of the Young modulus in an anisotropic zone on the radial stresses in it should be taken into account in the evaluation of radii of the plastic and transversally isotropic zones, which can form around the borehole. This can be fulfilled by employing the above-proposed approach to evaluate the effective Young modulus of a medium containing a system of oriented gas-filled fissures.
The complete system of equations for the problem of a borehole or a face in a rock mass with a cylindrical system of gas-filled fissures involves equilibrium equations, compatibility equations and constitutive equation (Hooke’s law).
Given the gas filtration is absent, the equilibrium equations are homogeneous, do not contain volume forces even at the existing pore-pressure gradient. For the case of plane strain state and the axial symmetry, the single equilibrium equation in the polar coordinates is:
0=−+rdr
d rr θσσσ. (30)
Here, rσ and θσ are the radial and peripheral stresses in the coal seam. Right now it is convenient to
express the peripheral stress in terms of the radial stress using equation (30):
rr r
drd σσσθ += . (31)
The single nontrivial compatibility equation for the plane deformation and axial symmetry is in the polar coordinates:
DEFORMATION OF A COAL SEAM WITH A SYSTEM OF ISOLATED GAS-FILLED FISSURES 35
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
0=−+rdr
d rεεε θθ . (32)
where rε , θε are radial and peripheral strains.
The stress-strain interaction obeys Hooke’s law for the transversally isotropic medium with the isotropy axis coinciding with the borehole radius. Hooke’s law for the studied 2D case in terms of (17) and (18) is:
θσσεEv
E rr
r −= 1,
(33)
θθ σσεEE
vr
1+−= ,
where rE is the effective Young modulus in the radial direction. The Poisson ratio corresponds to the
plane strain state, in publications these values differing from the respective values for the plane stress state and for 3D case are denoted by symbols with asterisks, but hereinafter this symbol is not used because the text does not contain magnitudes relating to the plane stress state. In general, these magnitudes can depend on actual and maximal stresses.
Using Eq. (18) for the deformation case in rocks containing isolated gas-filled fissures prone to propagation under the action of stresses additional to the initial compression stresses, expression (33) can be written as:
+−−+−+−= daaFapE
vdaaFaaap
Ev
Ev
E rrrrr )(][3
)1(16)()],,([
3)1(161
03
0
2
30
2
σσσσσε θ ,
(34)
θθ σσεEE
vr
1+−= .
Herein the stresses and strains are measured starting from the initial state, namely, the problem is solved in increments; Е, v are the effective Young modulus and Poisson ratio for rocks with the initial distribution of fissures )(0 aF of а diameter; 0p is the fissure pressure assumed equal in all the
fissures in the initial state; )(0 aF and )(aF are the initial and current distribution of fissures in rocks;
),,( 0 raap σ is the pressure in a fissure of the initial size 0a and the currently increased size а. If the
gas filtration is absent, the pore pressure is a function of the fissure size and the normal-to-fissure stress.
It is assumed in (34) that the single secondary tensile stress is the radial stress rσ , while the
peripheral stress θσ is the compression stress. Moreover, the influence of pressure variations in non-
growing fissures on the effective elastic characteristics is neglected in (34). It is also suggested that the effective elastic characteristics are sensibly affected by solely fissures oriented within a rather small angle, and their normal being very close to the actual radial stresses. Expression (34) differs from the analogous expression in [6] by the first equation where the last term accounts for the initial fissure distribution in rocks.
Expression (34) is simplified under the assumed substitution of one-size fissures for the fissure distribution member:
36 KOVALENKO et al.
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
NapE
vNaaap
Ev
Ev
E rrrrr300
23
0
2
][3
)1(16)],,([
3)1(161 +−−+−+−= σσσσσε θ , (35)
here, N is the number of fissures per unit volume, or the number of properly oriented fissures. Using (35) we can write the radial Young modulus rE as:
)(/ rr gEE σ= , (36)
where:
+−−
+−+= daaFa
pE
vdaaFaa
aaapE
vg
rr
r
rrr )(1
3)1(16
)(),(),),,((
13
)1(161)( 30
2
000300
2
σσσσσσ .(37)
Given the fissure density is high and makes it impossible to disregard their interaction, the differential scheme for calculation of efficient characteristics can be helpful. This procedure changes (37) for )( rg σ and leaves (36) the same. Using (36), we rewrite (33):
θσσσεEv
Eg
rr
r −= )(,
(38)
θθ σσεEE
vr
1+−= .
Therein, 1)( ≥rg σ is the function calculated from the analysis of the initial fissure distribution in
terms of the size, density of growth law of the fissures. The similar analysis was performed in [6] for the actual fissure distribution. The case when 1)( =rg σ corresponds to non-penetrating fissures.
As it is rather problematic to obtain the function )( rg σ in its accurate form, it is possible to resort
to some approximations, the simplest of which is the linear approximation, which can be built at two points: 1)( 0 =σg for an intact rock mass and Cg w =)(σ for a borehole contour point.
It is important to emphasize once more that the summarized account for the effect of the external stress and pressure in fissures by introducing effective characteristics (36) and (37) makes sense only when the deformation and the stress are measured from the initial magnitudes in the loaded rock mass as it is performed in [6]. In calculations of the respective zero stresses, the pressure-containing members in (34) should be considered as eigen, stress-free strains.
Substituting (31) into (38) and then into (32) gives equation for a relatively radial stress:
01)(3
22
2
=−−+ rrrr
rg
rdrd σσσσ
. (39)
Consider the elementary case with the approximation of 1)( −rg σ as a straight line:
0
)1(1)( σσσ r
r Ag −=− , (40)
where 0σ is the additional tensile stress applied to the borehole or face boundary; min/ EEA = is the
ratio of the initial effective modulus to the effective modulus under the applied stress. The numerical calculation results for 10=A are plotted as the shear stress distribution in Fig. 2. The solid lines correspond to the constant Young modulus, and dashed lines are the effective Young modulus in radial direction depending on the radial stress, thus at the accepted parameter values and the zero radial stress the magnitude of Young modulus falls 10 times as compared to the value at the initial stresses.
DEFORMATION OF A COAL SEAM WITH A SYSTEM OF ISOLATED GAS-FILLED FISSURES 37
JOURNAL OF MINING SCIENCE Vol. 48 No. 1 2012
Fig. 2. Stress distribution around a borehole.
It is reasonable to notice that the problem was solved by the above described algorithm for additional stresses, but the plots in Fig. 2 present the complete stresses.
It is pointed in Fig. 2 that the consideration for the reduction in the effective Young modulus in radial direction results in the increase in the shear stresses 2/)( θσσ −r in close vicinity of the
borehole and their reduction in the remote zone. It can be concluded, the consideration for the reduction in the effective Young modulus in the zone of deep, oriented gas-filled fissures in a rock mass leads to the reduction in a plastic zone adjacent to the free coal seam surface. According to the mechanism for the coal, rock or gas outburst origin, proposed in [4], the plastic zone functions as “a protective plug” hampering the outburst development, so any reduction in its size contributes to a greater outburst risk.
It is interesting to point out that the studied radial rock “weakening” increases stresses at the mine face contour and reduces the zone of its influence while it is known [17] that the effect of “rock weakening” in other directions is alternative: the stress concentration is reduced and the zone of the face influence grows with the feasible expansion of the maximum peripheral stress zone depthward the rock mass.
ACKNOWLEDGMENTS
The work was supported by the Russian Foundation for Basic Research, project, no. 08-08-00700-а.
REFERENCES
1. Khristianovich, S.A., “Free Soil Yield Induced by Expansion of Highly Pressurized Pore Gas. Crushing Wave,” Preprint no. 128, Moscow: IPM AN SSSR, 1979.
3. Khristianovich, S.A. and Salganik, R.L., “Outburst-Hazardous Situations. Crushing. Outburst Wave,” Preprint no. 152, Moscow: IPM AN SSSR, 1980.
4. Khristianovich, S.A., and Salganik, R.L., “Coal, Rock, and Gas Outbursts. Stresses and Strains,” Preprint no. 153, Moscow: IPM AN SSSR, 1980.
5. Salganik, R.L., “Effective Characteristics of a Material with Heavy Jointing. Geophysical Estimate of Jointing Parameters of a Seam in Terms of Outburst Prevention,” Preprint no. 154, Moscow: IPM AN SSSR, 1980.
7. Mokhel, A.N., “Theoretical Evaluation of a Protective Bed Influence on a Protected Bed,” Preprint no. 156, Moscow: IPM AN SSSR, 1980.
8. Kurlaev, A.R., “Stress-Strain State around a Cylindrical Face End under 3D Long-Distance Compression,” Preprint no. 158, Moscow: IPM AN SSSR, 1980.
9. Kurlaev, A.R., “Evaluation of Degassing Effect on a Free Flow of Soil with Pressurized Gas Content in Its Pores,” Preprint no. 163, Moscow: IPM AN SSSR, 1980.
10. Alekseev, A.D., Nedodaev, N.V., and Starikov, G.P., “Failure of Gas-Saturated Coal under Bulk Stress State in Unloading,” Preprint no. 139, Moscow: IPM AN SSSR, 1979.
11. Libovits, G. (Ed.), Razrushenie. Matematicheskie osnovy teorii razrusheniya (Failure. Mathematical Fundamentals of the Failure Theory), Moscow: Mir, 1975.
12. Vavakin, A.S. and Salganik, R.L., “Effective Elastic Characteristics of Bodies with Isolated Fissures, Voids, and Stiff Heterogeneities,” Izv. AN SSSR, Mekh. Tverd. Tela, 1978, no. 2.
13. Salganik, R.L., “Thin Elastic Layer Subjected to a Jump of Characteristics in an Infinite Elastic Body (Plane Body),” Izv. AN SSSR, Mekh. Tverd. Tela, 1977, no. 2.
14. Lekhnitsky, S.G., Teoriya uprugosti anozotropnogo tela (Theory of Elasticity of Anisotropic Body), Moscow: Nauka, 1977.
15. Salganik, R.L., “Mechanics of Bodies with Intensive Jointing,” Izv. AN SSSR, Mekh. Tverd. Tela, 1973, no. 4.
16. Shield, R.T., “Notes on Problems in Hexagonal Aelotropic Materials,” Proc. Cambridge Phill. Soc., 1951, vol. 47.
17. Baklashov, I.V. and Kartozia, B.A., Mekhanika podzemnykh sooruzhenii i konstruktsii krepei: Uchebnik dlya vuzov po spetsial’nosti “Shakhtnoe i podzemnoe stroitel’stvo” (Mechanics of Underground Structures and Support Systems: Textbook for Mine and Underground Construction), Moscow: Nedra, 1984.
